Download 7.1 Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . .

Algebra 1 Chapter 7 Note-Taking Guide
Factoring Polynomials
7.1
Name _______________________
Per ___ Date _________________
The whole numbers that are multiplied to find a product are called factors of that product. A number is
divisible by its factors. The order of factors does not change the product.
The circled factorization is the prime factorization because all the factors are prime numbers. The prime
factors can be written in any order, and except for changes in the order, there is only one way to write the
prime factorization of a number.
7.1
Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, . . .
7.1
Find the prime factorization of 48.
method 1: tree diagram
Find the prime factorization:
a.) 16
7.1
b.) 15
c.) 3
Factors that are shared by two or more whole numbers are called common factors. The greatest of these
common factors is called the greatest common factor, or GCF. We can use prime factorization to find
the GCF of any two terms.
d.) 12 and 32
e.) 24 and 18
f.) 15x3 and 9x2
7.1
method 2: reverse division
g.) 8x2y and 12x3y2
Helpful Hint: If two terms contain the same variable raised to different powers, the GCF will contain that
variable raised to the lower power.
h.) 36x2 and 45x3y2
i.) 12x and 28x3
7.2
Recall that the Distributive Property states that
ab + ac =a(b + c).
The Distributive Property allows you to “factor” out the GCF of the terms in a polynomial to write a
factored form of the polynomial.
Factor: 8x3 – 4x2 – 16x
7.2
b.) 8x4 + 4x3 – 2x2
c.) 12x3 + 6x2 – 10
d.) 15b + 9b3
e.) 5y – 12x2
Sometimes the GCF of terms is a binomial. This GCF is called a common binomial factor. You factor out
a common binomial factor the same way you factor out a monomial factor.
Factor:
f.) 5(x + 2) + 3x(x + 2)
g.) –2b(b2 + 1) + (b2 + 1)
You may be able to factor a polynomial by grouping. When a polynomial has four terms, you can make
two groups and factor out the GCF from each group.
h.) 6h4 – 4h3 + 12h – 8
i.) 5y4 – 15y3 + y2 – 3y
j.) 6b3 + 8b2 + 9b + 12
k.) 4r3 + 24r + r2 + 6
7.3
Factoring x2 + bx + c
a.) Factor x2 + 15x + 36
Multiply first and last terms. What are the factors of 36x2?
Which of the factors add to15x?
Check your answer:
7.3
b.) x2 + 10x + 24
c.) x2 + 6x + 5
d.) x2 – 8x + 15
e.) x2 – 5x + 6
f.) x2 – 3x – 18
g.) x2 + 2x – 15
h.) x2 – 8x – 20
i.) x2 – 13x – 48
7.4
Factoring ax2 + bx + c
When factoring trinomials with a number in front of the x2 makes factoring more complicated because we
now have to consider the factors of that number along with the factors of the constant.
Factor 6x2 + 11x + 4 and check your answer.
7.4
a.) 6x2 + 11x + 3
b.) 3x2 – 2x – 8
c.) 2x2 + 17x + 21
d.) 3x2 – 16x + 16
e.) 6x2 + 17x + 5
f.) 3n2 + 11n – 4
g.) 2x2 + 9x – 18
h.) 4x2 – 15x – 4
7.5
A trinomial is a perfect square if:

The first and last terms are perfect squares.

The middle term is two times the factor of the first term and the factor of the last term.
A perfect square trinomial is in the form of
2
2
2
 a + 2ab + b = (a+b)(a+b) or (a+b)

7.5
a2 - 2ab + b2 = (a-b)(a-b) or (a-b)2
Determine whether each trinomial is a perfect square. If so, factor. If not explain.
a.) 9x2 – 15x + 64
b.) 81x2 + 90x + 25
c.) x2 – 14x + 49
7.5
d.) 36x2 – 10x + 14
Difference of Squares Pattern:
a2 - b2 = (a + b)(a - b)
Factor each
e.) x2 - 9
g.) 81x4 - 121
f.) 4x2 - 1
h.) y6 – 169
7.6
We will now put together all of our factoring methods

common factors

difference of squares

perfect square trinomials

all other trinomial factoring
Factor 10x2 + 48x + 32
a.) 4x3 + 16x2 + 16x
b.) 2x2y – 2y3
c.) 9x2 + 3x – 2
d.) 12b3 + 48b2 + 48b
e.) 2x4 - 18
f.) x3 + 4x2 + 3x + 12