Unit 4: Polynomials
Terms:
Monomial: Polynomial with one term. Ex. x/2x/-5y/xy
Binomial: Polynomial with two terms. Ex. x+1/ 2x-y/abc-123
Trinomial: Polynomial with four terms.
N termed polynomial: Polynomial with n terms
Degree of a term: sum of exponents on the variables
Degree of a polynomial: highest degree of any terms
Like terms: terms that have the same variable factors. Ex. 2ab + 3ab
Multiplying Binomials
Steps:
1. Multiply the first term of each polynomial
2. Multiply the outside term of each polynomial
3. Multiply the inside term of each polynomial
4. Multiply the last term of each polynomial
5. Collect like terms
Also Known As… FOIL (First, outside, inside, last)
Example:
Squaring Binomials:
To square a binomial, use one of the following terms.
(a+b)2 = a2 + 2ab + b2 OR (a-b)2 = a2 – 2ab + b2
To find the product of the sum and difference of two terms (a difference of
squares) use the following pattern.
(a+b) (a-b) = a2 – b2
Factoring Polynomials working backwards from expanded form
Monomial common factor:
To factor this polynomial you need to
1. find the greatest common
factor (GCF) of the coefficients
2. find the greatest common
factor of the variable factors
3. “Factor out” the monomial
common factor
“Factor out” means to divide each
term by the MCF
Ex.
Factor:
8x3 – 6x2 + 4x2y
2x2 (4x-3y2+2y)
8x3 6x2y2 4x2y
2x2 2x2
2x2



The GCF of 8,6,4 is 2
The GCF of x3, x2, y2, and x2y
is x2
Our monomial common
factor is 2x2
Binomial common factor:
To factor this polynomial you need to
1. Identify the binomial that is
common to both portions of the
polynomial. This is the
binomial common factor
2. Factor out the binomial
common factor from each part
Ex.
2x(2+1) + 3y(2+1)
 Our binomial common
factor is (2+1)
= (2+1) (2x+3y)
Factoring By grouping:
To factor this polynomial you need to
1. Group terms that have a
common factor
2. Common monomial factor each
pair of terms
3. Look for a binomial common
factor and factor the
polynomial as outlined on the
previous instructions.
Ex. Factor
2m2-3t-6m+mt
=2m(m-3) + t (m-3)
= (2m + t) (m-3)
Factoring x2 + bx + c Trinomials:
To factor this polynomial you need to
1. Write x as the first term in each
binomial factor
2. Find two numbers that sum to
b and multiply to c
3. Use the numbers found in step
2 as the second terms in each
binomial factor
4. When you are factoring a
trinomial, ALWAYS common
factor first
Factoring x2 + bxy + cy2:
To factor this polynomial you need to
1. Write x as the first term in each
polynomial
2. Find two numbers that sum to
b and multiply to c
3. Use the first numbers in step
two with a y attached as the
second terms in each binomial
factor
4. When you are factoring a
trinomial, ALWAYS common
factor first.
Ex.
3x2 + 3x – 18
= 3(x2+x-6)
= 3(x+3) (x-2)
x2-2xy-15y2
= (x+3y) (x-5y)
Expanding AGAIN!!!
Factor by grouping
2x2 + 8x +3x + 12
= 2x2 + 11x + 12
= 2x (x+4) (x+3)  Grouping
= (x+4)(2x+3)  Binomial common factor
 to get the second step we broke up the 11x into 8x and 3x.
Factoring trinomials in the form ax2 +bx + c
1. Always common factor first if possible
2. Break up the middle term
- Replace the middle term (bx) by two terms whose coefficients have a sum of b
and a product of (a x c)
3. Factor by grouping
- Group pairs of terms and remove a common factor from each
- Then binomial common factor
Ex.
6x2 + 13 - 5
= 6x2 – 2x +15x -5
= 2x (3x-1) + 5(3x-1)
= (3x-1)(2x+5)
Factoring special quadratics
Ex.
9x2-16 9x2+Ox-16
= 9x2+12-12-16
=3x(3x+4)-4(3x+4)
= (3x+4)(3x-4)
Difference of squares
9x2-16
is a difference of squares!
To Factor a difference of squares use the formula
A2-b2=(a+b)(a-b)
Ex.
100p2-121q2
√
=10p
√
=11q
1002-1212
=(10p+11q)(10p-11q)
Summary
To factor a difference of squares:
1. Identify that both terms in your
binomial are perfect squares
2. Take the square of each term
3. Label roots as a+b
4. Factor a2-b2 as (a+b)(a-b)
To factor a perfect square trinomial:
1. Identify that the first and last
terms are perfect squares
2. Take the square root of the first
and last term
3. Label square root as a+b
4. Verify that the middle term is
equal to 2ab
5. Factor a2 +/- ab +b2 as (a+/- b)2
PRACTICE QUESTIONS:
Pg. 131-132 #’s 1-12, 13, 16, 19
Pg. 137 #’s 1-3, 4, 5
Pg. 137 #’s 8, 100
Pg. 143 #’s 1-8
Pg. 150 #’s 1-3, 4, 6
Pg. 156 #’s 1-6, 4
Pg. 163 #’s 1-5, 4, 5
Pg. 167 #’s 1-3, 6
Pg. 174-179 review again
Pg. 180-181 Chapter test