Download Ch 10 – Factoring 10.1 – Factors Factors: Prime Numbers

Ch 10 – Factoring
10.1 – Factors
Factors:
Prime Numbers:
Composite Numbers:
Neither:
Example: Find the factors of each number. Then classify each number as prime or composite.
47
25
35
23
Prime Factorization:
Example: Factor each monomial (prime factorization).
16b2c2
Greatest Common Factor (GCF):
9c3d
-15xy2
Examples: Find the GCF of each set of numbers or monomial.
12, 20, and 24
8 and 9
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.
10.2 – Factoring Using the Distributive Property
Factoring a Polynomial:
Example: 8y2 + 10y
Example: Factor each polynomial.
24y + 18y2
18fg – 21gh2
30x2 + 12x
15ab2 – 25abc
5a + 20ab + 10a2
17de – 15f
16a2b + 10ab2
20rs2 – 15r2s + 5rs
Find a missing factor:
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.
10.3 – F ctoring Trinomials: x2 + bx + c
Factoring Trinomials:
Example
x2 + x – 12
b2 + 4b + 4
x2 – 9x + 12
y2 – 7y + 12
x2 + 3x +2
n2 – 5n – 14
a2 + a + 3
m2 – m + 1
Factoring Trinomials:
Example
4x2 – 8x – 60
5m2 +45m + 100
3y2 – 9y – 54
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.
10.4 – Factoring Trinomials: ax2 + bx + c
Factoring Trinomials with a Leading Coefficient:
Examples:
2x2 – 7x + 3
5y2 + 2y – 3
2x2 – 9x + 4
3z2 – 8z + 4
3y2 + 7y – 6
4x2 – 4x – 15
2x2 + 3x + 1
6x2 + 17x + 5
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.
10.5 – Special Factors
Perfect Square Trinomials:
Example: Determine whether each trinomial is a perfect square trinomial. If so, factor it.
x2 + 14x + 49
a2 + 2a + 1
9a2 + 16a + 4
16x2 + 20x + 25
16b2 + 24b + 9
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:
Example: Determine whether each binomial is a difference of squares. If so, factor it.
d2 – 81
f2 + 64
4m2 – 144
Summary
121 – p2
25x3 – 100x
4a2 + 49