Download CH. 10

CH. 10
Factoring
10.1
Factors
Factors

Recall, the two numbers that are multiplied together
are called the factors of the product
3 x 4 = 12
Prime Numbers

Numbers that have exactly two factors, 1 and itself

Prime numbers less than 20:
 2,
3, 5, 7, 11, 13, 17, 19
Composite Numbers

Numbers that have more than two factors

Composite numbers less then 20:
 4,
6, 8, 9, 10, 12, 14, 15, 16, 18
Neither

0 and 1 are neither prime nor composite!!
Example

Find the factors of each number. Then classify each
number as prime or composite.
 47
 35
 25
 23
Prime Factorization


When a number is expressed as a product of prime
factors
Use a factor tree to find the prime factorization
12
Example

Factor each monomial (prime factorization).
16b2c2
9c3d
-15xy2
Greatest Common Factor

Two numbers may have common prime factors

The product of these prime factors is called the
greatest common factor

6 is the GCF of 36 and 42
Examples

Find the GCF of each set of numbers or monomials.
 12,
8
20, and 24
and 9
Examples

Find the GCF of each set of number or monomials.
 21ab2
and 9a2b
 24ab2c
and 60a2bc
Example

The area of a rectangle is 24 square inches. Find
the length and width so that the rectangle has the
least perimeter. Assume that the length and width
are both whole numbers.
Example

The area of a rectangle is 18 square inches. Find
the length and width so that the rectangle has the
least perimeter. Assume that the length and width
are both whole numbers.
Assignment

1st Assignment:
 P424:

1, 5 – 21
2nd Assignment:
 P424:
22 – 56 even, 58 – 68
10.2
Factoring Using the Distributive Property
Factoring

When you know the product and are asked to find
the factors

To factor means to write a polynomial as a product
of monomials and smaller polynomials
Factoring a Polynomial



Find the GCF
Write as a product of GCF and remaining factors
Use the distributive property
Example 
2
8y
+ 10y

Find the GCF

Write as a product of GCF and remaining factors

Use distributive property
Example

Factor each polynomial.
 24y
+ 18y2
 18fg
– 21gh2
Example

Factor each polynomial.
 30x2
+ 12x
 15ab2
– 25abc
Example

Factor each polynomial.
 5a
+ 20ab + 10a2
 17de
– 15f
Example

Factor each polynomial.
 16a2b
 20rs2
+ 10ab2
– 15r2s + 5rs
Finding a missing factor


If you know one of the factors, you can find the
other one by division
To divide a polynomial by a monomial, divide each
term of the polynomial by the monomial
Example

Divide (24a2 – 20a) by 4a.
Example

Divide (9b2 – 15) by 3.
Example

Divide (10x2y2 + 5xy) by 5xy.
Example

The diagram shows a walkway that is 2 meters wide
surrounding a rectangular planter. Write an
expression in factored form that represents the
area of the walkway.
Example

A stone walkway is to be built around a square
planter that contains a shade tree.
 If
the walkway is 2 meters wide, write an expression in
factored form that represents the area of the walkway.
Assignment

1st Assignment:
 P431:

2, 4 – 18
2nd Assignment:
 P432:
20 – 46 even, 47 – 57

P712: 10-2: 2 – 30 even
10.3
Factoring Trinomials: x2 + bx + c
Factoring Trinomials

One way is to factor a trinomial into two binomials
(x + 3)(x + 5) = x2 + 8x + 15
Example

x2 + x – 12

x2 – 9x + 12
Example

x2 + 3x +2

a2 + a + 3
Example

b2 + 4b + 4

y2 – 7y + 12
Example

n2 – 5n – 14

m2 – m + 1
Factoring Trinomials

If the trinomial has a leading coefficient, always
look for a GCF
Example

4x2 – 8x – 60

3y2 – 9y – 54
Example

5m2 +45m + 100

2x2 – 20x – 22
Example

Sahej is planning a rectangular garden in which the
width will be 2 feet less than the length. He will put
a composting box inside the garden that measures 2
feet by 4 feet. How many square feet are now left
for planting? Express the answer in factored form.
Example

Tammy is planning a rectangular garden in which the
width will be 4 feet less than its length. She has
decided to put a birdbath within the garden,
occupying a space 3 feet by 4 feet. How many
square feet are now left for planting? Express the
answer in factored form.
Assignment

1st Assignment:
 P438:

3 – 18
2nd Assignment:
 P438:
20 – 46 even, 49, 50, 53 - 62
10.4
Factoring Trinomials: ax2 + bx + c
Review
Factoring Trinomials with a Leading
Coefficient

If can’t factor out a GCF
 Use
reverse FOIL to factor into two binomials
Example

2x2 – 7x + 3
Example

2x2 – 9x + 4
Example

3y2 + 7y – 6
Example

2x2 + 3x + 1
Example

5y2 + 2y – 3
Example

3z2 – 8z + 4
Example

4x2 – 4x – 15
Example

6x2 + 17x + 5
Example

4x2 – 8x – 5
Example

The volume of a rectangular shipping crate is
2x3 – 4x2 – 30x. Find possible dimensions of
the crate.
Example

The volume of a rectangular shipping crate is
6x3 – 15x2 – 36x. Find possible dimensions for
the crate.
Assignment

1st Assignment
 P443:

3 – 12
2nd Assignment
 P443:
14 – 40 even, 41 – 43, 45 – 49, 52
10.5
Special Factors
Perfect Square Trinomials

The square of (x + 3) is the sum of
 The
square of the first term of the binomial
 The square of the last term of the binomial
 Twice the product of the terms of the binomial
Perfect Square Trinomials
Example

Determine whether each trinomial is a perfect
square trinomial. If so, factor it.
 x2 +
14x + 49
 9a2
+ 16a + 4
Example

Determine whether each trinomial is a perfect
square trinomial. If so, factor it.
 16b2
 a2
+ 24b + 9
+ 2a + 1
Example

Determine whether each trinomial is a perfect
square trinomial. If so, factor it.
 16x2
+ 20x + 25
 49x2
– 14x + 1
Example

The area of a square is d2 – 16d + 64. Find the
perimeter.
Example

The area of a square is x2 + 18x + 81. Find the
perimeter.
Factoring a Difference of Squares
Factoring a Difference of Two Squares
Example

Determine whether each binomial is a difference of
squares. If so, factor it.
 d2
– 81
 f2
+ 64
Example

Determine whether each binomial is a difference of
squares. If so, factor it.
 4m2
– 144
 121
– p2
Example

Determine whether each binomial is a difference of
squares. If so, factor it.
 25x3
 4a2
– 100x
+ 49
Summary
Assignment

1st Assignment
 P448:

3 – 12
2nd Assignment
 P448:
14 – 52 even, 54 – 62
Extra Practice P711:
 10-1:
1, 9, 11, 20 – 25
 10-2: 3 – 30 x 3s
 10-3: 2 – 14 even
 10-4: 2 – 16 even