Rational ExpressionsPolynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5,
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.PQ
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.PQ
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
All polynomials are rational expressions by viewing P as .P1
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.PQ
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
All polynomials are rational expressions by viewing P as .P1
x – 2 x2 – 2 x + 1 ,
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.PQ
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
All polynomials are rational expressions by viewing P as .P1
x – 2 x2 – 2 x + 1 ,
x(x – 2) (x + 1) (2x + 1) are rational expressions.
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.PQ
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
All polynomials are rational expressions by viewing P as .P1
x – 2 x2 – 2 x + 1 ,
x(x – 2) (x + 1) (2x + 1) are rational expressions.
x – 2 2 x + 1
is not a rational expression because the
denominator is not a polynomial.
Rational Expressions
For example, 7x12 – 4x5 + 6x2 – 5, πx – 0.762 are polynomials.
Rational (fractional) expressions are expressions of the
form , where P and Q are polynomials.PQ
Polynomials are expressions of the form anxn + an-1xn-1+ an-2xn-2+…+ a1x1+ a0 where the a’s are numbers.
All polynomials are rational expressions by viewing P as .P1
x – 2 x2 – 2 x + 1 ,
x(x – 2) (x + 1) (2x + 1) are rational expressions.
x – 2 2 x + 1
is not a rational expression because the
denominator is not a polynomial.
Rational expressions are expressions that describe calculation procedures that involve division (of polynomials).
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
The rational expression x2 – 4 x2 + 2x + 1 is in the expanded form.
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
The rational expression x2 – 4
(x + 2)(x – 2) (x + 1)(x + 1) .
is in the expanded form.
In the factored form, it’s
x2 + 2x + 1
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
The rational expression x2 – 4 x2 + 2x + 1 (x + 2)(x – 2) (x + 1)(x + 1) .
x – 2 x2 + 1 The expression is in both forms.
is in the expanded form.
In the factored form, it’s
Example A. Put the following expressions in the factored form.
a. x2 – 3x – 10 x2 – 3x
b. x2 – 3x + 10 x2 – 3
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
The rational expression x2 – 4 x2 + 2x + 1 (x + 2)(x – 2) (x + 1)(x + 1) .
x – 2 x2 + 1 The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a. x(x – 3)
x2 – 3x – 10 x2 – 3x
=
b. x2 – 3x + 10 x2 – 3
In the factored form, it’s
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
The rational expression x2 – 4 x2 + 2x + 1 (x + 2)(x – 2) (x + 1)(x + 1) .
x – 2 x2 + 1 The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a. x(x – 3)
x2 – 3x – 10 x2 – 3x
=
b. x2 – 3x + 10 x2 – 3
is in the factored form
In the factored form, it’s
Rational ExpressionsJust as polynomials may be given in the expanded form or factored form, rational expressions may also be given in these two forms.
The rational expression x2 – 4 x2 + 2x + 1 (x + 2)(x – 2) (x + 1)(x + 1) .
x – 2 x2 + 1 The expression is in both forms.
is in the expanded form.
Example A. Put the following expressions in the factored form.
(x – 5)(x + 2)a. x(x – 3)
x2 – 3x – 10 x2 – 3x
=
b. x2 – 3x + 10 x2 – 3
is in the factored form
Note that in b. the entire (x2 – 3x + 10) or (x2 – 3) are viewed as a single factors because they can’t be factored further.
In the factored form, it’s
We use the factored form to 1. solve equations 2. determine the domain of rational expressions
Rational Expressions
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs
Rational Expressions
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Rational Expressions
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations
Rational Expressions
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
PQ
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 + x2 – 2xx2 + 4x + 3 = 0
PQ
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3
x3 + x2 – 2xx2 + 4x + 3 = 0
Factor, we get
PQ
=
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3 = x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2xx2 + 4x + 3 = 0
Factor, we get
PQ
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3 = x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2xx2 + 4x + 3 = 0
Factor, we get
Hence for x3 + x2 – 2xx2 + 4x + 3 = 0, it must be that x(x + 2)(x – 1) = 0
PQ
We use the factored form to 1. solve equations 2. determine the domain of rational expressions2. evaluate given inputs3. determine of the signs of the outputs
Solutions of Equations
Example B. a. Write in the factored form and
solve the equation
The solutions of the equation of the form = 0 are the
zeroes of the numerator, so they are the solutions of P = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3 = x(x + 2)(x – 1)
(x + 3)(x + 1)
x3 + x2 – 2xx2 + 4x + 3 = 0
Factor, we get
Hence for x3 + x2 – 2xx2 + 4x + 3 = 0, it must be that x(x + 2)(x – 1) = 0
or that x = 0, –2, 1.
PQ
DomainThe domain of a formula is the set of all the numbers that we may use as input values for x.
Rational Expressions
DomainThe domain of a formula is the set of all the numbers that we may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
Rational Expressions
PQ
DomainThe domain of a formula is the set of all the numbers that we may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except where Q = 0.
Rational Expressions
PQ
PQ
Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except where Q = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
PQ
PQ
Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except where Q = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)(x + 3)(x + 1)
PQ
PQ
Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except where Q = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)(x + 3)(x + 1)
Hence we can’t have
PQ
PQ
(x + 3)(x + 1) = 0
Domain
b. Determine the domain of the formula
The domain of a formula is the set of all the numbers that we may use as input values for x. Because the denominator of a
fraction can’t be 0, therefore for the rational formulas
the zeroes of the denominator Q can’t be used as inputs.
in other words, the domain of are all the numbers except where Q = 0.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
x3 – 2x2 + 3xx2 + 4x + 3
Factor expression first.
= x(x + 2)(x – 1)(x + 3)(x + 1)
Hence we can’t have
PQ
PQ
(x + 3)(x + 1) = 0so that the domain is the set of all the numbers except –1 and –3.
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7,
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8)
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8)
3
4
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
Signs
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
d. Determine the signs (positive or negative) ofif x = ½,– 5/2 using the factored form.
x3 + x2 – 2xx2 + 4x + 3
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
d. Determine the signs (positive or negative) ofif x = ½,– 5/2 using the factored form.
x3 + x2 – 2xx2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs of each factor,
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
d. Determine the signs (positive or negative) ofif x = ½,– 5/2 using the factored form.
x3 + x2 – 2xx2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs of each factor, we get +( + )( – )
(+)(+)
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
d. Determine the signs (positive or negative) ofif x = ½,– 5/2 using the factored form.
x3 + x2 – 2xx2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs of each factor, we get = –, so it’s negative.+( + )( – )
(+)(+)
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
d. Determine the signs (positive or negative) ofif x = ½,– 5/2 using the factored form.
x3 + x2 – 2xx2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs of each factor, we get
Plug in x = –5/2 into the factored form we get –( – )( – )(+)( – )
= –, so it’s negative.+( + )( – )(+)(+)
Evaluation
c. Evaluate if x = 7 using the factored form.
It is often easier to evaluate expressions in the factored form.
Rational Expressions
x3 + x2 – 2xx2 + 4x + 3
Using the factored form x(x + 2)(x – 1)(x + 3)(x + 1)
Plug in x = 7, we get 7(9)(6)(10)(8) =
3
418940
We use the factored form to determine the sign of an output.Signs
d. Determine the signs (positive or negative) ofif x = ½,– 5/2 using the factored form.
x3 + x2 – 2xx2 + 4x + 3
Plug in x = ½ into the above factored form and check the signs of each factor, we get = –, so it’s negative.+( + )( – )
(+)(+)Plug in x = –5/2 into the factored form we get –( – )( – )
(+)( – ) = +so that the output is positive.
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.Example B. Reduce the following expressions.
(x – 2)(x + 3)(x + 3)(x + 2)a.
b. x2 – 3x + 10 x2 – 3
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.Example B. Reduce the following expressions.
(x – 2)(x + 3)
It's already factored, proceed to cancel the common factor.(x + 3)(x + 2)
(x – 2)(x + 3)(x + 3)(x + 2)
a.
b. x2 – 3x + 10 x2 – 3
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.Example B. Reduce the following expressions.
(x – 2)(x + 3)
1 = x – 2
x + 2
It's already factored, proceed to cancel the common factor.(x + 3)(x + 2)
(x – 2)(x + 3)(x + 3)(x + 2) which is reduced.
a.
b. x2 – 3x + 10 x2 – 3
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.Example B. Reduce the following expressions.
(x – 2)(x + 3)
1 = x – 2
x + 2
It's already factored, proceed to cancel the common factor.(x + 3)(x + 2)
(x – 2)(x + 3)(x + 3)(x + 2) which is reduced.
a.
b. x2 – 3x + 10 x2 – 3
This is in the factored form.
PNQN = =
1
Rational ExpressionsCancellation Law: Common factor may be cancelled, i.e.
PNQN
PQ
A rational expression is reduced (simplified) if all common factors are cancelled.To reduce (simplify) a rational expression, put the expression in the factored form then cancel the common factors.Example B. Reduce the following expressions.
(x – 2)(x + 3)
1 = x – 2
x + 2
It's already factored, proceed to cancel the common factor.(x + 3)(x + 2)
(x – 2)(x + 3)(x + 3)(x + 2) which is reduced.
a.
b. x2 – 3x + 10 x2 – 3
This is in the factored form. There are no common factors so it’s already reduced.
Rational Expressionsc. x2 – 1
x2 – 3x+ 2x2 – 1
x2 – 3x+ 2= (x – 1)(x + 1) (x – 1)(x – 2)Factor then cancel
Rational Expressionsc. x2 – 1
x2 – 3x+ 2x2 – 1
x2 – 3x+ 2= (x – 1)(x + 1) (x – 1)(x – 2)
1
Factor then cancel
Rational Expressionsc. x2 – 1
x2 – 3x+ 2x2 – 1
x2 – 3x+ 2= (x – 1)(x + 1) (x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Rational Expressions
Only factors may be canceled.
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite:
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2Cancellation of Opposite Factors
The opposite of a quantity x is the –x.
While identical factors cancel to be 1, opposite factors cancel to be –1,
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Rational Expressions
Only factors may be canceled. It is wrong to cancel terms, i.e. P + Q P + R = P + Q
P + R
For example, x2 + 1 x2 – 2 = x2 + 1
x2 – 2
The opposite of a quantity x is the –x.
While identical factors cancel to be 1, opposite factors cancel to be –1, in symbol,
original: y -z x – y v – 4u – 2w
opposite: -y z -x + y or y – x -v + 4u + 2w or …
x –x = –
1.
c. x2 – 1 x2 – 3x+ 2
x2 – 1 x2 – 3x+ 2= (x – 1)(x + 1)
(x – 1)(x – 2)
1
= x + 1 x – 2Factor then cancel
Cancellation of Opposite Factors
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y)
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1= –(2 + x)
x + 1 or –2 – x x + 1
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1= –(2 + x)
x + 1 or –2 – x x + 1
In the case of polynomials in one variable x, if the highestdegree term is negative, we may factor out the negative sign then factor the expressions.
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1= –(2 + x)
x + 1 or –2 – x x + 1
–x2 + 4–x2 + x + 2
Example D. Pull out the “–” first then reduce.
In the case of polynomials in one variable x, if the highestdegree term is negative, we may factor out the negative sign then factor the expressions.
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1= –(2 + x)
x + 1 or –2 – x x + 1
In the case of polynomials in one variable x, if the highestdegree term is negative, we may factor out the negative sign then factor the expressions.
–x2 + 4–x2 + x + 2 =
Example D. Pull out the “–” first then reduce. –(x2 – 4)
–(x2 – x – 2)
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1= –(2 + x)
x + 1 or –2 – x x + 1
–x2 + 4–x2 + x + 2 = (x – 2)(x + 2)
(x + 1)(x – 2)=
Example D. Pull out the “–” first then reduce. –(x2 – 4)
–(x2 – x – 2)
In the case of polynomials in one variable x, if the highestdegree term is negative, we may factor out the negative sign then factor the expressions.
b(x – y)a(y – x)
Example C.
2y–2y =
-1–1
-1= –b
a
a.
b.
Rational Expressions
(x – 2y)(x + 3y) (x – 3y)(2y – x)
c.-1
= –(x + 3y) (x – 3y) or –x – 3y
x – 3y
d. 4 – x2
x2 – x – 2 = (2 – x)(2 + x)(x + 1)(x – 2)
-1= –(2 + x)
x + 1 or –2 – x x + 1
–x2 + 4–x2 + x + 2 = (x – 2)(x + 2)
(x + 1)(x – 2)=
Example D. Pull out the “–” first then reduce. –(x2 – 4)
–(x2 – x – 2) = x + 2 x + 1
In the case of polynomials in one variable x, if the highestdegree term is negative, we may factor out the negative sign then factor the expressions.
Rational ExpressionsTo summarize, a rational expression is reduced (simplified) if all common factors are cancelled.Following are the steps for reducing a rational expression.1. Factor the top and bottom completely. (If present, factor the “ – ” from the leading term)2. Cancel the common factors: -cancel identical factors to be 1 -cancel opposite factors to be –1
Ex. A. Write the following expressions in factored form. List all the distinct factors of the numerator and the denominator of each expression.
1.
Rational Expressions
2x + 3 x + 3 2. 4x + 6
2x + 6 3. x2 – 4 2x + 4
4. x2 + 4x2 + 4x 5. x2 – 2x – 3
x2 + 4x 6. x3 – 2x2 – 8xx2 + 2x – 3
7. Find the zeroes and list the domain of x2 – 2x – 3x2 + 4x
8. Use the factored form to evaluate x2 – 2x – 3x2 + 4x
with x = 7, ½, – ½, 1/3.
9. Determine the signs of the outputs of x2 – 2x – 3x2 + 4x
with x = 4, –2, 1/7, 1.23.
For problems 10, 11, and 12, answer the same questions
as problems 7, 8 and 9 with the formula .x3 – 2x2 – 8xx2 + 2x – 3
Ex. B. Reduce the following expressions. If it’s already reduced, state this. Make sure you do not cancel any terms and make sure that you look for the opposite cancellation.
13.
Rational Expressions
2x + 3 x + 3
20. 4x + 6 2x + 3
22. 23. 24.
21.
3x – 12x – 4
12 – 3xx – 4
4x + 6 –2x – 3
3x + 12x – 4
25. 4x – 6 –2x – 3
14. x + 3 x – 3 15. x + 3
–x – 3
16. x + 3x – 3
17. x – 3 3 – x 18. 2x – 1
1 + 2x
19. 2x – 1 1 – 2x
26. (2x – y)(x – 2y)(2y + x)(y – 2x) 27. (3y + x)(3x –y)
(y – 3x)(–x – 3y)
28. (2u + v – w)(2v – u – 2w)(u – 2v + 2w)(–2u – v – w) 29. (a + 4b – c)(a – b – c)
(c – a – 4b)(a + b + c)
30.
Rational Expressions
37.
x2 – 1x2 + 2x – 3
36. 38. x – x2
39. x2 – 3x – 4
31. 32.
33. 34. 35.
40. 41. x3 – 16x
x2 + 4 2x + 4
x2 – 4x + 4
x2 – 4x2– 2x
x2 – 9x2 + 4x + 3
x2 – 4 2x + 4
x2 + 3x + 2x2 – x – 2 x2 + x – 2
x2 – x – 6x2 – 5x + 6
x2 – x – 2x2 + x – 2
x2 – 5x – 6
x2 + 5x – 6x2 + 5x + 6
x3 – 8x2 – 20x
46.45. 47. 9 – x2
42. 43. 44. x2 – 2x9 – x2
x2 + 4x + 3– x2 – x + 2
x3 – x2 – 6x–1 + x2
–x2 + x + 2x2 – x – 2
– x2 + 5x – 61 – x2
x2 + 5x – 6
49.48. 50.xy – 2y + x2 – 2xx2 – y2 x3 – 100x
x2 – 4xy + x – 4yx2 – 3xy – 4y2
Ex. C. Reduce the following expressions. If it’s already reduced, state this. Make sure you do not cancel any terms and make sure that you look for the opposite cancellation.