Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.

Section 5.1

Quadratic Equations

OBJECTIVES

A Find the greatest common factor (GCF) of numbers.

B Find the GCF of terms.

OBJECTIVES

C Factor out the GCF.

D Factor a four-term expression by grouping.

DEFINITIONGreatest Common Factor (GCF)

The largest common factor of the integers in a list.

PROCEDUREFinding the Product

4(x + y) = 4x + 4y

5(a – 2b) = 5a – 10b

2x(x + 3) = 2x2 + 6x

PROCEDUREFinding the Factors

4x + 4y = 4(x + y)

5a – 10b = 5(a – 2b)

2x2 + 6x = 2x(x + 3)

DEFINITIONGCF of a Polynomial

The term axn is the GCF of a polynomial if

1. a is the greatest integer that divides each coefficient.

DEFINITIONGCF of a Polynomial

The term axn is the GCF of a polynomial if

2. n is the smallest exponent of x in all the terms.

Section 5.1Exercise #2

Chapter 5Factoring

18 30 2

9 15 3

3 5

GCF = 2 • 3 = 6

Find the GCF of 18x2y4 and 30x3y5 .

x2y4

x3y5

x2y4

GCF = 6x2y4


Chapter 5Factoring

Factor 2x3 + 6x2y + x + 3y.

Factor out GCF

(3) Terms(2) Terms (4) Terms

Difference of Squares

Sum/Difference of Cubes

Perfect Square Trinomial

(x2 + bx + c)

(ax2 + bx + c)

Grouping

Factoring Strategy Flow Chart

Factor 2x3 + 6x2y + x + 3y.

3 2 = 2 + 6 + + 3x x y x y

2 = 2 + 3 + 1 + 3x x y x y

2 = + 3 2 + 1x y x

Section 5.2

Quadratic Equations

OBJECTIVES

A Factor trinomials of the form x 2 + bx + c.

RULEFactoring Rule 1

X 2 + (A + B)X + AB = (X + A)(X + B)

(F1)

PROCEDUREFactoring x2 + bx + cFind two integers whose product is c and whose sum is b.

1. If b and c are positive, both integers must be positive.


2. If c is positive and b is negative, both integers must be negative.


3. If c is negative, one integer must be negative and one positive.


Chapter 5Factoring

Factor x2 – 8x + 12.

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor x2 – 8x + 12.

= – 6 – 2x x

Section 5.3

Quadratic Equations

OBJECTIVES

A Use the ac test to determine whether ax 2 + bx + c is factorable.

OBJECTIVES

B

Factor ax2 + bx + c by grouping.

OBJECTIVES

C

Factor ax2 + bx + c using FOIL.

TEST

ac test for ax2 + bx + c

A trinomial of the form ax2 + bx + c is factorable if there are two integers with product ac and sum b.

TESTac test

ax2 + bx + c

We need two numbers whose product is ac.

The sum of the numbers must be b.

PROCEDUREFactoring by FOIL

ax2 + bx + c = (__x + __)(__x + __)

Product must be c.

Product must be a.

PROCEDURE

1. The product of the numbers in the first (F) blanks must be a.

Factoring by FOIL

PROCEDUREFactoring by FOIL

2. The coefficients of the outside (O) products and the inside (I) products must add up to b.

PROCEDURE

3. The product of numbers in the last (L) blanks must be c.

Factoring by FOIL


Chapter 5Factoring

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor 6x2 – 11xy + 3y2 .

3 2 x x – y – 3y

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Section 5.4

Quadratic Equations

OBJECTIVES

A Recognize the square of a binomial (a perfect square trinomial).

OBJECTIVES

B Factor a perfect square trinomial.

OBJECTIVES

C Factor the difference of two squares.

RULESFactoring Rules 2 and 3: PERFECT SQUARE TRINOMIALS

X 2 + 2AX + A2 = (X + A)2 (F2)

22 2Note that + ( + )X A X A

RULESFactoring Rules 2 and 3: PERFECT SQUARE TRINOMIALS

X 2 – 2AX + A2 = (X – A)2

22 2Note that – ( – )X A X A (F3)

RULEFactoring Rule 4: THE DIFFERENCE OF TWO SQUARES

X2– A2 = (X + A)(X – A) (F4)


Chapter 5Factoring


Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping



= 3x – 2y 2


Chapter 5Factoring

Factor 16x2 – 25y2 .

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor 16x2 – 25y2 .

= 4 + 5 4 – 5x y x y

Section 5.5

Quadratic Equations

OBJECTIVES

A Factor the sum or difference of two cubes.

OBJECTIVES

B Factor a polynomial by using the general factoring strategy.

OBJECTIVES

C Factor expressions whose leading coefficient is –1.

RULEFactoring Rule 5: THE SUM OF TWO CUBES.

X 3+ A3 = (X + A)(X 2 – AX + A2)

(F5)

RULEFactoring Rule 6: THE DIFFERENCE OF TWO CUBES. (F6)

X 3– A3 = (X – A)(X 2+ AX + A2)

PROCEDUREGeneral Factoring Strategy

1. Factor out all common factors.


2. Look at the number of terms inside the parentheses. If there are:

Four terms: Factor by grouping.


Three terms:

If the expression is a perfect square trinomial, factor it. Otherwise, use the ac test to factor.


Two terms and squared:

Look at the difference of two squares (X

2–A2) and factor it.

Note: X 2+A2 is not factorable.


Two terms and cubed:

Look for the sum of two cubes (X

3+A3) or the difference of two cubes (X

3-A3) and factor it.


Make sure your expression is completely factored.

Check by multiplying the factors you obtain.

Chapter 5Factoring

Section 5.5


Chapter 5Factoring

Factor 8y3 – 125x3 .

3 3 = 2 – 5y x

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor 8y3 – 125x3 .

= 2y 3 – 5x 3

= 2y – 5x 4y2 + 10xy + 25x2


Chapter 5Factoring

Factor 2x3 – 8x2 – 10x.

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor 2x3 – 8x2 – 10x.

2 = 2 – 4 – 5x x x

= 2 – 5 + 1x x x

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor 2x3 + 6x2 + x + 3.

2 = + 3 2 + 1x x

2 = 2 + 3 + 1 + 3x x x

3 2 = 2 + 6 + + 3x x x


Chapter 5Factoring

Factor – 9x4 + 36x2 .

Factor out GCF





(x2 + bx + c)

(ax2 + bx + c)

Grouping


Factor – 9x4 + 36x2 .

2 2 = – 9 – 4x x

2 = – 9 + 2 – 2x x x

Section 5.6

Quadratic Equations

OBJECTIVES

A Solve quadratic equations by factoring.

DEFINITIONQuadratic Equation in Standard Form

If , and are real numbers (a 0),

a b c

ax2+ bx + c = 0

PROCEDURESolving Quadratics by Factoring

1. Perform necessary operations on both sides so that right side = 0.


2. Use general factoring strategy to factor the left side if necessary.


3. Use the principle of zero product and make each factor on the left equal 0.


4. Solve each of the resulting equations.


5. Check results by substituting solutions obtained in step 4 in original equation.


Chapter 5Factoring

(2x – 3)(x – 4) = 2(x – 1) – 1Solve.

2x2 – 8x – 3x + 12 = 2x – 2 – 1

2x2 – 11x + 12 = 2x – 3

2x2 – 13x + 12 = – 3

2x2 – 13x + 15 = 0

2x – 3 = 0 x – 5 = 0or

(2x – 3)(x – 5) = 0

(2x – 3)(x – 4) = 2(x – 1) – 1Solve.

2x – 3 = 0 x – 5 = 0or

2x = 3 x = 5

x =

32

Section 5.7

Quadratic Equations

OBJECTIVESA Integer problems.

B Area and perimeter problems.

OBJECTIVESC Problems involving the

Pythagorean Theorem.

D Motion problems.

NOTETerminology

Examples: 3,4; – 6,–5

Notation2 consecutive integers

n, n+1

NOTETerminology

Examples: 7, 8, 9; – 4,– 3,– 2

Notation3 consecutive integers

n, n+1, n+2

NOTETerminology

Examples: 8,10; – 6,– 4

Notation2 consecutive even integers

n, n +2

NOTETerminology

Examples: 13,15; – 21,– 19

Notation2 consecutive odd integers

n, n +2

DEFINITIONPythagorean Theorem

a2 + b2 = c2

If the longest side of a right triangle is of length c and the other two sides are of length a and b, then

DEFINITIONPythagorean Theorem

a2 + b2 = c2

Leg a

Hypotenuse c

Leg b


Chapter 5Factoring

The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.

Let x + 2 = 2nd odd integer

x (x + 2) = 10(x + 2) + 13

x2 + 2x = 10x + 20 + 13

x2 + 2x = 10x + 33

x2 – 8x = 33

x2 – 8x – 33 = 0

Let x = 1st odd integer


x2 – 8x – 33 = 0

(x – 11)(x + 3) = 0

x – 11 = 0 x + 3 = 0 or

x = 11 x = – 3 or




x = 11 x = – 3 or

x + 2 = 13 x + 2 = – 1

The integers are 11 and 13 or – 3 and – 1.




Chapter 5Factoring

A rectangular 10-inch television screen (measured diagonally) is 2 inches wider than it is high. What are the dimensions of the screen?

x2 + (x + 2)2 = 102

Let x = height

x

Let x + 2 = length

x + 2

10

x2 + (x + 2)2 = 102

x 2 + x2 + 4x + 4 = 100

2x2 + 4x + 4 = 100

2x2 + 4x – 96 = 0

x2 + 2x – 48 = 0

(x + 8)(x – 6) = 0

x + 8 = 0 x – 6 = 0 or x = – 8 x = 6

height –8 x + 2 = 8

The screen is 6 inches high and 8 inches long.

Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.

Documents

c slide

x slide

factoring slide

foil slide

form slide

objectives b slide

procedure factoring

bx c ax