9.1 – ADDING AND SUBTRACTING POLYNOMIALS
Definitions
monomial – an expression that is a number, variable, or a product of a
number and one or more variables
degree of a monomial – the sum of the exponents of its variables
Example 1
Find the degree of each monomial.
a.
1
- x
2
b. 7x2y3
c. -14
Definitions
polynomial – a monomial or the sum or difference of two or more
monomials
standard form of a polynomial – the degrees of its monomial terms
decrease from left to right
degree of a polynomial – the degree of the monomial with the greatest
exponent
Example: 3x4 + 5x2 – 7x + 1
Naming a
Polynomial
Polynomial
Degree
Name
Using
Degree
Number of
Terms
7x + 4
3x + 2x + 1
1
2
Linear
Quadratic
2
3
Name
Using
Number of
Terms
Binomial
Trinomial
4x3
9x4 + 11x
3
4
1
2
Monomial
Binomial
5
0
Cubic
Fourth
degree
Constant
1
Monomial
2
Example 2
Write each polynomial in standard form. Then name each polynomial
based on its degree and the number of its terms.
a. 6x2 + 7 – 9x4
b. 3y – 4 – y3
c.
Example 3
8 + 7v – 11v
You can add polynomials by adding like terms.
Simplify the following.
a. (12m2 + 4) + (8m2 + 5)
b. (t2 – 6) + (3t2 + 11)
c.
(9w3 + 8w2) + (7w3 + 4)
d. (2p3 + 6p2 + 10p) + (9p3 + 11p2 + 3p)
Example 4
Subtracting polynomials will follow the same process as adding, with
one additional step.
Remember to distribute the subtraction to each value in the second
polynomial – “keep, change, change.” Then, we can add just like we
did in the last example.
Simplify the following.
a. (v3 + 6v2 – v) – (9v3 – 7v2 + 3v)
b. (30d3 – 29d2 – 3d) – (2d3 + d2)
c. (4x2 + 5x + 1) – (6x2 + x + 8)
9.2 – MULTIPLYING AND FACTORING
Methods
Distributive Property
- we will use the distributive property to multiply by a monomial
Example 1
Simplify each product.
a. 4b(5b2 + b + 6)
b. -7h(3h2 – 8h – 1)
c.
Example 2
2x(x2 – 6x + 5)
Factoring a polynomial reverses the multiplication process. To begin
factoring, we first find the greatest common factor (GCF) of its terms.
Find the GCF of the terms of each polynomial.
a. 5v5 + 10v3
b. 3t2 – 18
c.
4b3 – 2b2 – 6b
Example 3
Factor each polynomial.
a. 8x2 – 12x
b. 5d3 + 10d
c.
6m3 – 12m2 – 24m
9.3 – MULTIPLYING BINOMIALS
Methods
There are two methods to multiply binomials.
1. FOIL
2. The Box
Example 2
Simplify each product.
* FOIL – First, Outer, Inner, Last
a. (x + 7)(x + 9)
b. (3x – 4)(2x + 5)
c.
(y + 4)(5y – 8)
d. (7m – 2p)2
Example 3
Simplify each product.
* The Box
a. (a – 8)(a – 9)
b. (3c – 7)(2c – 5)
c. (5y + 1)2
d. (2y + 5)(y – 3)
Example 4
Find the area of each shaded region. Simplify.
a.
2x - 6
x+3
b.
x
3x -2
Example 5
You may also be asked to multiply a binomial and a trinomial.
Simplify each product.
a. (6n – 8)(2n2 + n + 7)
b. (4x2 + x – 6)(2x – 3)
9.5 – FACTORING TRINOMIALS OF THE TYPE x2 + bx + c
Definitions
Factoring – reversing the process of multiplication
15 = 3 x 5
We will use the box and work in reverse to find the binomials that
multiplied to reach the trinomial.
Example 1
Factor each expression.
a. g2 + 7g + 10
b. v2 + 21v + 20
c.
k2 – 10k + 25
d. q2 – 15q + 36
e. m2 + 8m – 20
f.
y2 – y – 56
Example 2
You can also factor trinomials that have more than one variable.
Factor each expression.
a. x2 + 11xy + 24y2
b. v2 + 2vw – 48w2
9.6 – FACTORING TRINOMIALS OF THE TYPE ax2 + bx + c
Note
We will continue to use the box and work backwards to find the
binomials that multiplied to give the trinomial result.
Example 1
Factor each expression.
a. 2y2 + 5y + 2
b. 6n2 – 23n + 7
c. 2y2 – 5y + 2
d. 5d2 – 14d – 3
e. 2n2 + n – 3
f. 20p2 – 31p – 9
Example 2
Some polynomials have a GCF, always check for a greatest common
factor before trying to factor the problem. The smaller the numbers
are the easier factoring will be for you!
Factor each expression.
a. 2v2 – 12v + 10
b.
4y2 + 14y + 6
c. 18k2 – 12k – 6
Examples
Factor each expression.
a. x2 + 8x + 16
b. n2 – 16n + 64
c. 9g2 – 12g + 4
d. 4t2 + 36t + 81
Example 2
Factor each expression.
a. x2 – 36
b. m2 – 100
c.
9v2 – 4
d. 4w2 – 49
Example 3
Remember to look for a GCF 1st !
Factor each expression.
a. 8y2 – 50
b. 28k2 – 7