Topic 1Common Factors

Unit 7 Topic 1

Explore1. Multiply the factors to find the matching expanded form. 

Try this on your own first!!!!

Explore1. Multiply the factors to find the matching expanded form. 

2 2 3

2 6

x

x

2

2 2 3

2 6

xx x

x x

2

3

3

xx x

x x

3

1

2

Explore2. Factor the expanded form to find the matching factors.

2 2

( 3)2

x x

x x 3

2

1

( 6)

x x

x x

2 2

( 6)2

x x

x x

InformationAn algebraic term is a number, a variable or a product of a number and a variable. For example, 3, and are terms. The polynomial is formed when you add or subtract terms. • A polynomial with one term is called a monomial.

For example, 7, , , and are all monomials.

• A polynomial with two terms is called a binomial. For example, , are binomials.

The product of a monomial and a binomial can be determined using the distributive property. The distributive property is a rule that states .

2x 37x

24x 2x y6x

4x 3 5x

( )a b c ab ac

Information

A factor is any number or variable that when multiplied with one or more other numbers or variables forms a product. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12. The factors of are .

The greatest common factor (GCF) is the largest factor shared by two or more terms. For example, the GCF of 12 and 18 is 6. The GCF of and is .

In order to factor a polynomial, the first step is always to determine the greatest common factor (if there is one). Once determined, it can be factored out (divided out) of all the terms.

2x y , and x x y

2x y 2 3x y 2x y

212 18 6 (2 3)x x x x

Example 1Multiplying Using the Distributive Property

Use the distributive property to expand each of the following. a)

b)

c)

Try this on your own first!!!!Try this on your own first!!!!

22( 3 5)

(4 7)

3 (5 4)

x x

x x

x x

Example 1: Solutions

a) b)2

2

2

2 2 3 5

2

2

( 3 5)

6 10

2

x x

x

x

x

x

2

(4

4

7

7)

7

4

xx

x

x

x

x x

Example 1: Solutions

c)

2

5 4

15 12

3 3

3 (5 4)

x x

x

x

x

x

x

Example 2Identifying the Greatest Common Factor

Identify the GCF of each of the following.a)

b)

c)

d)

Try this on your own first!!!!

2

2

2

16 and 24

and x

6 and 9x

2 , 6 and -10

x

x

x x

Example 2: Solutionsa)

b)

c)

d)

2

2

2

2

16 and 24

and

6 and 9

2 , 6 and -1

3

2

8

0

x x

x x

x

x

x

x

Example 3Writing a Polynomial in Factored Form Using the GCF

Factor each of the following.

a)

b)

Try this on your own first!!!!

2

2

3

4 6

x x

x x

Example 3: Solutions

a) b)2

( 3)

3

x x

x

x

x

x

2

(2 3

2

2

6

)

2

4

x x

x

x

x

x

Example 4Writing Formulas in Factored Form Using the GCF

The formula for the surface area of a right cylinder is . Express in factored form.

Try this on your own first!!!!

22 2SA r rh

Example 4: Solutions

22 2

2

2 (

2

)

r rhS

S

r

rA r

r

h

A

More Information

A quadratic equation is an equation of the form . The solutions to a quadratic equation are called the roots of the equation.

Some quadratic equations can be solved by factoring. This approach depends on the zero product rule.

Zero Product RuleIf AB = 0 then A = 0, or B = 0, or both.

2 0ax bx c

Example 5Solving Using the Zero Product Rule

Find the roots to the following quadratic equations.

a) b)

c) d)

Try this on your own first!!!!

2 3 0

5 2 3 0

x

x x

7 0

3 4 1 0

x x

x x

Example 5: Solutionsa) b)

c) d)

2 3 0

5 2 3 0

3 0

3

5 0 2 3 0

0 2 3

32

x

x

x x

x x

x

x

x x

0 7 0

7

3 0 4 1 0

7 0

3

0 4 1

1

1

4

4 0

x x

x

x x

x x

x

x x

x x

Example 6Solving Quadratic Equations Using Common Factoring

Factor and then solve each of the following quadratic equations.a)

b)

Try this on your own first!!!!

2

5 15 0

7 0

x

x x

Example 6: Solutions

a) b)

( 3

5 15

) 0

3 0

5 50

3

5 x

x

x

x

2

( 7) 0

0 7 0

,

0

7

0 7

7x x

x

x

x

x

x

x x

x

Example 6: Solutionsc) d)

2

( 4

3 12

) 0

3

0

0 4 0

0, 4

3 3

3 x

x x

x

x xx x

x

2

(2 3) 0

2 0 2 3 0

4 6

30,

20

2

2

2 x

x x

x

xx

x

xx

Example 6: Solutionse)

2

(2 3)

12 1

0

6 0 2 3 0

3

6

0,

6

0

2

86x

x

x

x

x

x

x

x

x

Example 7Determining the Width of the Gateway Arch

The Gateway Arch in St. Louis, Missouri, is the tallest man-made monument in the Western Hemisphere. The height, h, in metres, of a point on the arch is approximately modelled by the formula , where d is the horizontal distance from one end of the arch, in metres. a) At both ends of the arch the height is 0 m. What equation would you use to determine the width of the arch?

Try this on your own first!!!!

20.02 4h d d

Example 7Determining the Width of the Gateway Arch

a) At both ends of the arch the height is 0 m. What equation would you use to determine the width of the arch?

20.02 40 d d

Example 7: Solution

b) Determine the approximate width of the Gateway Arch.

The width is the difference between the two horizontal distances, 200 - 0 = 200 m, so the width is 200m.

2

00.02 0.02

00 ( 200)

0 0.0

0.02 4

2 0 200

0 0

2

2

.0

0

d d

d d

d d

d d

d d

Need to Know:• Factoring is the reverse of expanding or

multiplying.

• To find the greatest common factor (GCF) of a polynomial, find the GCF of the coefficients and variables.

• To factor a GCF from a polynomial divide each term by the GCF.

Need to Know:• Polynomials can be written as a product of the GCF

and the sum or difference of the remaining terms.

• You can solve some quadratic equations by factoring. First write the equation in the form , with one side of the equation equal to zero. Then factor the other side. Next, set each factor to zero, and solve for the unknown.

You’re ready! Try the homework from this section.

26 15 3 2 5x x x x

2 0ax bx c