Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Name/Period: _______________________________________
Unit Calendar
Date
Mar. 4
Mar. 8
Mar. 10
Mar. 14
Mar. 16
Mar. 18
Mar. 29
Sect.
9.1
Topic
Homework
HW √ On-Time
Adding & Subtracting
Page 557: 3-26 odd
Polynomials
9.2
Multiplying & Factoring Page 565: 3-35 odd
Essential Question: How do we determine which numbers we can add
together/subtract from one another?
9.3
Multiplying Binomials
Page 572: 3, 5, 7, 11, 13,
Factoring
15, 23-33 odd
Essential Question: What property of numbers do we apply when multiplying
binomials?
9.5
Quiz #1
Page 586: 3-17
9.6
Factoring
Page 596: 4-21
Essential Question: How do you determine the factors of a quadratic equation?
9.7
Factoring
Page 603: 3-19
9.8
Page 610: 3-11
Essential Question: How do you determine the factors of a quadratic equation?
Dividing Polynomials
Page 788: 3-18
Essential Question: What do division and factoring have in common?
Quiz #2
Review Packet
Unit Review
Study, study, study
Essential Question: How is factoring and simplifying radicals similar?
Unit Test
Essential Question: How do you determine the factors polynomial equations?
Page 1 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Page 2 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Vocabulary
Monomial – a real number, a variable, or a product of a real number and one or more variables with
whole number exponents. Examples of monomials:
18
-4x2
z
2.5xy3
a/3
Degree of a monomial – the sum of the exponents of its variables.
5x
Degree is 1 (The exponent is 1).
6x3y2
Degree is 5 (The exponents are 3 and 2).
4
Degree is 0 (4 = 4x0; the degree of a nonzero constant is 0).
Polynomial – a monomial or sum of monomials. For example, 3x4 + 5x2 – 7x + 1 is a polynomial.
Standard form of a polynomial – the degrees of its monomial terms decrease from left to right.
5x3 + 10x2 – 13x + 22 is written in standard form
10x + 5x4 – 7x2 + 1 is not in standard form
Degree of a polynomial – the degree of the monomial with the greatest exponent. For example, the
degree of 3x4 + 5x2 – 7x + 1 is 4.
Names of Polynomials:
Monomial
–
polynomial with one term
–
5x5
Binomial
-
polynomial with two terms
-
5x + 9
Trinomial
-
polynomial with three terms -
4x2 + 7x + 3
Adding & Subtracting Polynomials
You add and subtract polynomials by combining like terms. You can use one of two methods:
Method 1: Add vertically – Line up like terms. Then add the coefficients.
Method 2: Add horizontally – Group like terms. Then add the coefficients.
Page 3 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Examples. A researcher studied the number of overnight stays in the U.S. National Park Service
campgrounds and in the backcountry of the national park system over a 5-year period. The researcher
model the results, in thousands, with the following polynomials.
-7.1x2 – 180x + 5800
Campgrounds:
Backcountry:
21x2 – 140x + 1900
In each polynomial, x = 0, corresponds to the first year in the 5-year period. What polynomial models
the total number of overnight stays in both campgrounds and backcountry?
Method 1
Add vertically
+
-7.1x2 – 180x + 5800
21x2 – 140x + 1900
13.9x2 – 320x + 7700
Method 2
Add horizontally
(-7.1x2 – 180x + 5800) + (21x2 – 140x + 1900)
(-7.1x2 + 21x2) + (-180x – 140x) + (5800 + 1900)
13.9x2 – 320x + 7700
A polynomial that models the number of stays (in thousands) in campgrounds and backcountry of the
5-year period is 13.9x2 – 320x + 7700.
What is a simpler form of (x3 – 3x2 + 5x) – (7x3 + 5x2 – 12)?
Method 1
Subtract vertically
-
x3 – 3x2 + 5x
(7x3 + 5x2
– 12)
x3 – 3x2 + 5x
-7x3 – 5x2
+ 12
Need to distribute!!!
-6x3 – 8x2 + 5x +12
Method 2
Subtract horizontally
(x3 – 3x2 + 5x) – (7x3 + 5x2 – 12)
x3 – 3x2 + 5x – 7x3 - 5x2 + 12 Need to distribute!!!
(x3 – 7x3) + (-3x2 – 5x2) + 5x + 12
-6x3 – 8x2 + 5x +12
Page 4 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Practice. Simplify each sum or difference.
(-12x3 + 106x2 – 241x +4477) + (14x2 – 14x +1545)
(-4m3 – m + 9) – (4m2 + m -12)
(x + 7x2) - (1 + 3x + x2)
(7 + 2x – 4x2) + (-3x + x2 – 5)
Page 5 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Multiplying & Factoring
You can use the Distributive Property to multiply a monomial by a polynomial.
Examples.
2x(3x + 1)
= 2x(3x) + 2x(1)
-x3(9x4 – 2x3 - 7)
=
= 6x2 + 2x
5n(3n3 – n2 + 8)
=
4x(3x2 – 2x + 6)
=
-3x(5x – 8)
=
(6n – 7)(5n3)
=
(7x2 + x – 3)(-2x2)
=
6x(2X3 + 7x)
=
4xy(2x + 7y)
=
x2y(3x2 – 9y)
=
Practice.
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Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring a polynomial reverses the multiplication process. When factoring a monomial from a
polynomial from a polynomial, the first step is to find the greatest common factor (GCF) of the
polynomial’s terms.
 Find the GCF and then reverse distribute.
What is the GCF of 5x3 + 25x2 + 45x?
What is the GCF of 3x4 – 9x2 – 12x?
Once you find the GCF, you can factor it out of the polynomial.
Example. What is the factored form of 4x5 – 24x3 + 8x?
1) Find the GCF.
2) “Reverse distribute.”
What is the factored form of 9x6 + 15x4 + 12x2?
Page 7 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Using the Graphing Calculator to find Greatest Common Factors
1) Press the [MATH] key
2) Arrow over to NUM
3) Arrow down (or press 9) to number “9: gcd(“ and press [ENTER]
4) Enter “first number” “comma” “second number” then [ENTER]
(Note: The numbers you are entering are the coefficients of the variables in the polynomial)
5) The answer is the greatest common factor of the numbers.
Practice. Factor each expression.
14x4 – 21x2
2x2 + 8
12x2 + 27x
-8x3 – 16
33x5 – 121x2
30x4 + 45x3 – 105x
18x3y + 24x2y2 – 42xy3
6x5 + 9x
15m7n3 – 50m2n3
210x6yz2 – 360x3yz4
Page 8 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Multiplying Binomials
You can use the distributive property to find the product of two binomials. Consider the first binomial
as a single variable and distribute it to each term of the first binomial.
Example. What is the simpler form of (3x – 7)(2x + 4)?
Distribute the first factor, 3x – 7.
(3x – 7)(2x + 4)
=
2x(3x – 7) + 4(3x – 7)
Distribute 2x
Distribute 4
Combine like terms
Distribution works well to multiply a binomial by a trinomial.
What is the simpler form of (x + 3)(2x2 + x + 4)?
Page 9 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Alternatively, you can use FOIL to multiply binomials. FOIL is an alternative to distribution when
working with binomials.
F
O
I
L
-
First Outer Inner Last -
Multiply each of the first terms in the binomials.
Multiply each of the outer terms in the binomials.
Multiply each of the inner terms in the binomials.
Multiply each of the last terms in the binomials.
Example. What is the simpler form of (5x – 3)(2x +1)?
(5x – 3)(2x +1) =
First
(5x)(2x)
+
Outer
(5x)(1)
+
Inner
(-3)(2x)
+
Last
(-3)(1)
Multiply
Combine like terms
What is the simpler form of (3x – 4)(x + 2)?
Page 10 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Practice. What is the simpler form of each product? Use either distribution or FOIL.
(3x – 4)(x + 2)
(a + 4)(a2 + 3 – 2a)
(2x – 5)(3x2 + x – 1)
(x + 8)(x + 5)
(x + 3)(x – 5)
(x – 3)(x – 7)
(2x – 7)(3x2 + x – 5)
(2p2 + 3)(2p – 5)
Page 11 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring ax2 + bx + c, when a = 1
Standard form for a quadratic equation is ax2 + bx + c
a, b, and c are real numbers x is a variable
To factor, you find the two binomials that you could multiply to get the same quadratic equation.
Factoring is “reverse-FOILing”. You end up with two binomials instead of starting with them.
When a = 1, the factors will be (x +/- [real number 1])(x +/- [real number 2])
The sum of the two real numbers is b.
* real number 1 + real number 2 = b *
The product of the two real numbers is c.
* real number 1 x real number 2 = c *
Example. What is the factored form of x2 + 8x + 15?
a=1
x2 + 8x + 15 = (x + 5)(x + 3)
b=8
c = 15
5+3=8
5 x 3 = 15
Check using FOIL:
(x)(x) + (3)(x) + (5)(x) + (3)(5)
=
x2 + 3x + 5x + 15
=
x2 + 8x + 15
If c is positive, the 2 real numbers will have the same sign.
If b is positive, the numbers will both be positive.
(x + [real number 1])(x + [real number 2])
If b is negative, the numbers will both be negative.
(x - [real number 1])(x - [real number 2])
Page 12 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Examples.
b is POSITIVE
b is NEGATIVE
x2 + 3x + 2 = (x + 2)(x + 1)
x2 - 5x + 6 = (x - 2)(x - 3)
x2 + 7x + 12 = (x + 3)(x + ____)
x2 - 8x + 12 = (x - 6)(x - ____)
x2 + 9x + 20 = (x + ____)(x + ____)
x2 - 10x + 16 = (x - _____)(x - ____)
If c is negative, the 2 real numbers will have different signs.
If b is positive, the number farthest from zero will be positive.
If |real number 1| > |real number 2|
(x + [real number 1])(x - [real number 2])
If b is negative, the number farthest from zero will be negative.
If |real number 1| > |real number 2|
(x + [real number 1])(x - [real number 2])
Examples.
b is POSITIVE
b is NEGATIVE
x2 + x - 6 = (x + 3)(x - 2)
x2 - 2x - 8 = (x + 2)(x - 4)
x2 + 3x + 10 = (x + 5)(x - ____)
x2 - 7x - 8 = (x + 1)(x - ____)
x2 + 12x - 13 = (x + ____)(x - ____)
x2 - 4x - 8 = (x + _____)(x - ____)
Page 13 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
𝑏
𝑏
𝑏
𝑎∙𝑐
𝑎∙𝑐
𝑎∙𝑐
Page 14 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
𝑏
𝑏
𝑏
𝑎∙𝑐
𝑎∙𝑐
𝑎∙𝑐
Page 15 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Practice. What is the factored form of each polynomial?
r2 + 11r + 24
x2 – 11x + 24
n2 + 9n – 36
c2 – 4c – 21
x2 – 2x – 35
x2 + 6x – 55
Page 16 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring ax2 + bx = c when a > 1
You can write some trinomials of the form ax2 + bx = c as the product of two binomials.
Example. Factor the trinomial 6x2 + 23x + 7
We are able to “break apart” and rewrite equations as long as the value remains constant. Therefore,
we can rewrite the trinomial to be:
Rewrite:
6x2 + 2x + 21x + 7
Factor out GCF:
2x(3x + 1) + 7(3x + 1)
Distributive Property:
(2x + 7)(3x + 1)
23x = 2x + 21x
Why can we do this? How does it work? What??????
Reversing the Distributive Property – it’s like factoring.
2n + 7n = n(2 + 7)
= 9n
3a – 4ab = a(
5x + 10 = 5(x + 2)
)
6xy + 2y =
We can use an expression (usually a binomial) instead of a variable.
4(x + 3) + x(x + 3) = (x + 3)(4 + x)
(2n + 5)3 – (2n + 5)n = (2n+ 5)(3 – n)
9(n- 3) – 2n(n – 3) =
7(x – 2) + x(x – 2) =
Page 17 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring when a > 1
Examples. What is the factored form of each trinomial?
2x2 + 11x + 5
3x2 + 5x + 2
8x2 – 14x – 15
𝑏
𝑏
𝑏
𝑎∙𝑐
𝑎∙𝑐
𝑎∙𝑐
Page 18 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
2x2 + 21x – 11
𝑏
𝑏
𝑏
𝑎∙𝑐
𝑎∙𝑐
𝑎∙𝑐
Page 19 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Practice.
4x2 + 27x + 35
2x2 + 3x – 9
6x2 – 21x – 9
The area of a rectangle is 2x2 – 13x – 7. What are the possible dimensions of the rectangle? Use
factoring.
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Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring Special Cases
Perfect-square Trinomials: The result of squaring a binomial.
x2 + 8x + 16
=
4n2 – 12n + 9 =
(x + 4)(x + 4)
=
(x + 4)2
(2n – 3)(2n – 3)
=
(2n – 3)2
How to recognize a perfect-square trinomial.

The first and last terms are perfect squares.
Example: x2 + 8x + 16
The first term, x2, is a perfect square of x.
The last term, 16, is a perfect square of 4.

The middle term is twice the product of one factor from the first term and one factor from the
second term.
Example: x2 + 8x + 16
The middle term, 8x, is twice the product of x and 4.
𝟐 ∙ 𝟒 ∙ 𝒙 = 𝟖𝒙
Let’s verify 16x2 – 56x + 49
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Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Example. What is the factored form of x2 -12x + 36?
Practice. What is the factored form of each trinomial?
x2 + 6x +9
4n2 – 12n + 9
Page 22 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring a Difference of Two Squares
For all real numbers, a and b:
a2 – b2 = (a + b)(a – b)
Examples:
x2 – 64 = (x + 8)(x – 8)
25x2 – 36 = (5x + 6)(5x – 6)
What is the factored form of x2 – 9?
Practice. What is the factored form of each of the following expressions.
v2 – 100
s2 – 16
16x2 – 81
25d2 – 64
Page 23 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Factoring by Grouping
You can factor higher degree polynomials by using grouping.
Example. What is the factored form of 3n3 – 12n2 + 2n – 8?
Try grouping the two terms with the highest degrees together.
(3n3 – 12n2) + (2n – 8)
Factor out the GCF of each group of two terms.
3n2(n – 4) + 2(n – 4)
Factor out the common factor, n – 4
(3n2 + 2)(n – 4)
Check by FOIL.
What is the factored form of 8t3 + 14t2 + 20t + 35?
4q4 – 8q3 + 12q2 – 24q
Practice. What is the factored form of 6h4 + 9h3 +12h2 +18h?
Page 24 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Division of Polynomials & Prime Polynomials
Dividing a polynomial by a monomial is very similar to factoring (finding the GCF).
Examples.
8𝑥 3 −6𝑥 2
(25𝑥 2 + 10𝑥) ÷ 5𝑥
2𝑥 2
8𝑥 3
6𝑥 2
= 2𝑥 2 − 2𝑥 2
= 5𝑥 2 + 2
= 4𝑥 − 3
3xy 6𝑥 2 𝑦 + 3𝑥𝑦 − 12𝑥𝑦 2
= 2𝑥 + 1 − 4𝑦
Practice. You can check your answers by multiplying the answer by the divisor (re-distribute)!
(𝟐𝟒𝒙𝟒
𝟑
− 𝟏𝟔𝒙 + 𝟖𝒙
𝟐
𝟐)
÷ 𝟖𝒙
𝒙𝟑 −𝟒𝒙
𝟐
𝟕𝒂𝒃 𝟒𝟐𝒂 𝒃 + 𝟏𝟒𝒂𝒃 − 𝟐𝟏𝒂𝒃
𝒙
𝟐
(−𝟑𝟑 + 𝟒𝟒𝒙 − 𝟓𝟓𝒙𝟐 ) ÷ (−𝟏𝟏𝒙)
𝟏𝟖𝒎𝟓 +𝟒𝟓𝒎𝟒 −𝟑𝟔𝒎𝟑
𝟗𝒎𝟑
𝟓𝟔𝒎𝟕 𝒏𝟑 −𝟏𝟔𝒎𝟒 𝒏𝟒 −𝟐𝟒𝒎𝟐 𝒏𝟓
−𝟖𝒎𝒏𝟑
Page 25 of 26
Briggs – Algebra 1 – Unit 8: Factoring and Polynomial Division -- Notes
Prime Polynomials.
A polynomial is prime if it cannot be factored.
The standard form of a polynomial (quadratic equation) is ax2 + bx + c = 0
The discriminant is b2 – 4ac.
If b2 – 4ac is a positive number, it can be factored.
If b2 – 4ac is a negative number, it cannot be factored.
Prime!
Are these prime?
x2 + 6x – 7
2x2 + 6x – 5
x2 – 4x + 4
3x2 – 4x – 6
w2 – 6w + 16
x2 – 11x + 24
y2 + 19y + 60
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