tiplication and Division of Rational Express Frank Ma © 2011
Multiplication and Division of Rational Expressions
Frank Ma © 2011
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 )
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2)
(x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
Multiplication Rule for Rational Expressions A B
C D
* =
AC BD
Multiplication and Division of Rational Expressions
In most problems, we reduce the product by factoring the top and the bottom, then cancel.Example A. Simplify
10xy3z
a. * y2
5x3 =
10xy2
5x3y3z =
2x2yz
b. (x2 + 2x – 3 ) (x – 2) (x2 – x )
(x2 – 4 )
=(x + 3)(x + 2)
x
*
=
(x + 3)(x – 1 ) (x – 2) x(x – 1 )
(x + 2 )(x – 2)
In the next section, we meet the following type of problems.
Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.
a. x + 3 x – 1 (x2 – 1)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1)
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) – x + 1
(x – 3)(x + 1)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
Example B. Simplify and expand the answers.
Multiplication and Division of Rational Expressions
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
= x2 – x – 2 – x2 – 4x – 3
Example B. Simplify and expand the answers.
Multiplication and Division of Rational ExpressionsExample B. Simplify and expand the answers.
a. x + 3 x – 1 (x2 – 1)
=
x + 3 (x – 1) (x – 1)(x + 1)
=
(x + 3)(x + 1) = x2 + 4x + 3
b. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
=
x – 2 (x – 3)(x + 3) [ – x + 1
(x – 3)(x + 1) ] ( x – 3)(x + 3)(x + 1) (x + 1) (x + 3)
=
(x – 2)(x + 1) – (x + 1)(x + 3)
= (x – 2)(x + 1) + (–x –1)(x + 3)
= x2 – x – 2 – x2 – 4x – 3
= –5x – 5
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
We convert division by an expression of multiplying by its reciprocal.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)*
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)*
=2(x – 3) (x + 3)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
Example C. Simplify (2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)*
(9 – x2)
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)*
(–1)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Division Rule for Rational ExpressionsMultiplication and Division of Rational Expressions
AB
CD
÷ = ADBC
Reciprocate
(2x – 6) (x + 3) ÷ (x2 + 2x – 3)
(9 – x2)
=(2x – 6) (x + 3)
(x2 + 2x – 3) (9 – x2)*
=2(x – 3) (x + 3)
(x + 3)(x – 1) (3 – x)(3 + x)*
(–1)
=–2(x – 1)
(3 + x)
Example C. Simplify
We convert division by an expression of multiplying by its reciprocal. Then we factor and reduce the product.
Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences.
Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this.
Multiplication and Division of Rational ExpressionsBesides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(2x – 6) 3x2
b. =
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. =
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2 = – 2
x22
3x
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2 = – 2
x22
3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable.
Multiplication and Division of Rational Expressions
Example D. Break up the numerators as the sums or differences and simplify each term.
(2x – 6) (x + 3)
a. =
Besides the expanded form and factored forms, rational expressions may also be split into sums or differences. There are two common ways to do this. I. Split off the numerator term by term.
(x + 3) – (x + 3)2x 6
(2x – 6) 3x2
b. = – 2x 63x2 3x2 = – 2
x22
3xII. Long DivisionLong division is the extension of the long division of numbers from grade school and it is for the division of polynomials in one variable. Specifically, long division gives relevant results only when the degree of the numerator is the same or more than the degree of the denominator.
Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient.
Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
1
Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
1
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
Multiplication and Division of Rational ExpressionsLet’s look at the example 125/8 or 125 ÷ 8 by long division.i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
1
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered.
Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered.
405
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R
Dand that R (the remainder) is smaller then D (no more quotient).
405
where Q is the quotient
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and move from left to right until there is enough to enter a quotient. )8 125
15
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
845
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R
Dand that R (the remainder) is smaller then D (no more quotient).
405
1258 = 15 + 5
8
where Q is the quotient
Let’s look at the example 125/8 or 125 ÷ 8 by long division.
Multiplication and Division of Rational ExpressionsExample E. Divide using long division(2x – 6)
(x + 3)
Multiplication and Division of Rational Expressions
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Make sure the terms are in order.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Make sure the terms are in order.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
enter the quotients of the leading terms 2x/x = 2
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
enter the quotients of the leading terms 2x/x = 2
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
2x + 6ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
–)
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered.
–)
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
–)
Stop. No more quotient since x can’t going into 12.iii. Repeat steps i and ii until no more
quotient may be entered.
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)
Hence we may write(2x – 6) (x + 3)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)
Hence we may write(2x – 6) (x + 3)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Q
R
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational Expressions
)x + 3 2x – 6 2
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
2x + 6–12
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
= 2 – 12 x + 3
–)
Hence we may write(2x – 6) (x + 3)
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Q
R
Q R
Example E. Divide using long division(2x – 6) (x + 3)
Multiplication and Division of Rational ExpressionsExample F. Divide using long division
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3x – 2
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Multiplication and Division of Rational Expressions
)x + 3
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
Make sure the terms are in order.
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27–)30
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient
–)x2 – 6x + 3
–9x – 27–)30
Stop. No more quotient since x can’t going into 30. Hence 30 is the remainder.
and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27–)30
Hence x2 – 6x + 3
x – 2=
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
Example F. Divide using long divisionx2 – 6x + 3x – 2
Multiplication and Division of Rational Expressions
)x + 3 x – 9
ii. Multiply the quotient back into the problem and subtract the results from the numerator. Bring down the remaining terms from the numerator.
x2 + 3x–9x + 3
iii. Repeat steps i and ii until no more quotient may be entered. Then we may put the fraction N/D in the following mixed form:
–)x2 – 6x + 3
–9x – 27–)30
Hence x2 – 6x + 3
x – 2= x – 9 + 30
x + 3
i. Put the problem in the long division format with the “bottom-out” and enter the quotients of the leading terms.
ND = Q + R
Dhas smaller degree then denominator D (no more quotient).
where Q is the quotient and the remainder R
Example F. Divide using long divisionx2 – 6x + 3x – 2
Ex A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions
1. 10x * 25x3
15x4
* 1625x4
10x* 35x32.
5. 109x4
* 185x3
6.
3.12x6 * 56x14
56x6
27* 63
8x5
10x* 35x34.
7. 75x49
* 4225x3
8.
9. 2x – 4 2x + 4
5x + 10 3x – 6
10. 6 – 4x 3x – 2
x – 2 2x + 4
11. 9x – 12 2x – 4
2 – x 8 – 6x
12. x + 4 –x – 4
4 – xx – 4
13. 3x – 9 15x – 5
3 – x 5 – 15x
14. 42 – 6x –2x + 14
4 – 2x –7x + 14
*
*
*
*
*
*
15.(x2 + x – 2 ) (x – 2) (x2 – x)
(x2 – 4 )*
16. (x2 + 2x – 3 ) (x2 – 9) (x2 – x – 2 )
(x2 – 2x – 3)*
17.(x2 – x – 2 ) (x2 – 1) (x2 + 2x + 1)
(x2 + x )*
18. (x2 + 5x – 6 ) (x2 + 5x + 6) (x2 – 5x – 6 )
(x2 – 5x + 6)*
19.(x2 – 3x – 4 ) (x2 – 1) (x2 – 2x – 8)
(x2 – 3x + 2)*
20.(– x2 + 6 – x ) (x2 + 5x + 6) (x2 – x – 12 )
(6 – x2 – x)*
Ex. A. Simplify. Do not expand the results. Multiplication and Division of Rational Expressions
21. (2x2 + x – 1 ) (1 – 2x)
(4x2 – 1) (2x2 – x )
22. (3x2 – 2x – 1) (1 – 9x2)
(x2 + x – 2 ) (x2 + 4x + 4)
23.(3x2 – x – 2) (x2 – x + 2) (3x2 + 4x + 1)
(–x – 3x2)24. (x + 1 – 6x2)
(–x2 – 4)(2x2 + x – 1 ) (x2 – 5x – 6)
25. (x3 – 4x) (–x2 + 4x – 4)
(x2 + 2) (–x + 2)
26. (–x3 + 9x ) (x2 + 6x + 9)(x2 + 3x) (–3x2 – 9x)
Ex. B. Multiply, expand and simplify the results.
÷
÷
÷
÷
÷
÷
27. x + 3 x + 1 (x2 – 1) 28. x – 3
x – 2 (x2 – 4) 29. 2x + 3 1 – x (x2 – 1)
30. 3 – 2x x + 2 (x + 2)(x +1) 31. 3 – 2x
2x – 1 (3x + 2)(1 – 2x)
32. x – 2 x – 3 ( + x + 1
x + 3 )( x – 3)(x + 3)
33. 2x – 1 x + 2 ( – x + 2
2x – 3 ) ( 2x – 3)(x + 2)
Multiplication and Division of Rational Expressions
38. x – 2 x2 – 9 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 3)(x + 1)
39. x + 3 x2 – 4 ( – 2x + 1
x2 + x – 2 ) ( x – 2)(x + 2)(x – 1)
40. x – 1 x2 – x – 6 ( – x + 1
x2 – 2x – 3 ) ( x – 3)(x + 2)(x + 1)
41. x + 2 x2 – 4x +3( – 2x + 1
x2 + 2x – 3 ) ( x – 3)(x + 3)(x – 1)
34. 4 – x x – 3 ( – x – 1
2x + 3 )( x – 3)(2x + 3)
35. 3 – x x + 2 ( – 2x + 3
x – 3 )(x – 3)(x + 2)
Ex B. Multiply, expand and simplify the results.
36. 3 – 4x x + 1 ( – 1 – 2x
x + 3 )( x + 3)(x + 1)
37. 5x – 7 x + 5 ( – 4 – 5x
x – 3 )(x – 3)(x + 5)
Ex. C. Break up the following expressions as sums and differences of fractions.
42. 43. 44.
45. 46. 47.
x2 + 4x – 6 2x2x2 – 4
x2
12x3 – 9x2 + 6x3x
x2 – 4 2x
xx8 – x6 – x4
x2x8 – x6 – x4
Ex D. Use long division and write each rational expression in
the form of Q + .RD
(x2 + x – 2 ) (x – 1)
(3x2 – 3x – 2 ) (x + 2)
2x + 6 x + 2 48. 3x – 5
x – 2 49. 4x + 3 x – 1 50.
5x – 4 x – 3 51. 3x + 8
2 – x52. –4x – 5 1 – x53.
54. (2x2 + x – 3 ) (x – 2)
55. 56.
(–x2 + 4x – 3 ) (x – 3)
(5x2 – 1 ) (x – 4)
57. (4x2 + 2 ) (x + 3)
58. 59.
Multiplication and Division of Rational Expressions