ALGEBRA III

Polynomials

Quick Review

Addition, Subtraction and Multiplication

Polynomials

A term can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables.

A term that is a product of constants and/or variables is called a monomial.

Examples of monomials: 8, w, 24 x3y

A polynomial is a monomial or a sum of monomials.

Examples of polynomials: 5w + 8, 3x2 + x + 4, x, 0, 75y6

A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Consider the chart below with examples of different types of polynomials.

Monomials Binomials Trinomials Polynomials

5x2 3x + 4 3x2 + 5x + 9 5x3 6x2 + 2xy 9

8 4a5 + 7bc 7x7 9z3 + 5 a4 + 2a3 a2 + 7a 2

8a23b3 10x3 7 6x2 4x ½ 6x6 4x5 + 2x4 x3 + 3x 2

Definitions Dealing with Polynomials

The degree of a term of a polynomial is the number of variable factors in that term. The degree of 9x5 is 5.

The part of a term that is a constant factor is the coefficient of that term. The coefficient of 4y is 4.

The leading term of a polynomial is the term of highest degree. Its coefficient is called the leading coefficient and its degree is referred to as the degree of the polynomial. Consider this polynomial below:

4x2 9x3 + 6x4 + 8x 7

The terms are 4x2, 9x3, 6x4, 8x, and 7.The coefficients are 4, 9, 6, 8 and 7.

The degree of each term is 2, 3, 4, 1, and 0.

The leading term is 6x4 and the leading coefficient is 6.

The degree of the polynomial is 4.

Operations on Polynomials

Polynomials can be added and subtracted by combining like terms (using the Distributive Property).

Polynomials are customarily written in descending order, starting with the term with the highest power.

Example Set No.1

Combine like terms and write in descending order.

a) 4y4 9y4

b) 7x5 + 9 + 3x2 + 6x2 13 6x5

c) 9w5 7w3 + 11w5 + 2w3

Example Set No. 1 Solutionsa) 4y4 9y4 = (4 9)y4 = 5y4

b) 7x5 + 9 + 3x2 + 6x2 13 6x5

= 7x5 6x5 + 3x2 + 6x2 + 9 13

= x5 + 9x2 4

c) 9w5 7w3 + 11w5 + 2w3

= 9w5 + 11w5 7w3 + 2w3

= 20w5 5w3

Example Set No.2 To add or subtract polynomials, perform the operation on like terms only. Simplify each expression.

(a) (6x3 + 7x 2) + (5x3 + 4x2 + 3)

(b) (8x4 x3 + 9x2 2x + 72)

Example Set No. 2 Solutions a) (6x3 + 7x 2) + (5x3 + 4x2 + 3)

= (6 + 5)x3 + 4x2 + 7x + (2 + 3)

= x3 + 4x2 + 7x + 1

b) (8x4 x3 + 9x2 2x + 72)

= 8x4 + x3 9x2 + 2x 72

Example Set No.3

Simplify each expression. Write in descending order.

a) (6x2 4x + 7) (10x2 6x 4)

b) (10x5 + 2x3 3x2 + 5) (3x5 + 2x4 5x3 4x2)

Example Set No. 3 Solutions

a) 6x2 4x + 7 You can do you work (10x2 6x 4) in column form if it’s 4x2 + 2x + 11 easier.

b) (10x5 + 2x3 3x2 + 5) (3x5 + 2x4 5x3 4x2)

= 10x5 + 2x3 3x2 + 5 + 3x5 2x4 + 5x3 + 4x2

= 13x5 2x4 + 7x3 + x2 + 5

Multiplying Polynomials

When multiplying a monomial by a polynomial, use the Distributive Law.

5x2(x3 4x2 + 3x 5)

= 5x5 20x4 + 15x3 25x2

When multiplying two binomials, use the process referred to as FOIL.

(A + B)(C + D) = AC + AD + BC + BD

Multiply First terms: AC.

Multiply Outer terms: AD.

Multiply Inner terms: BC

Multiply Last terms: BD

↓

FOIL

(A + B)(C + D)

OI

F

L

Example Set No.4

Find each product.

a) (x + 8)(x + 5)

b) (y + 4) (y 3)

c) (5t3 + 4t)(2t2 1)

d) (4 3x)(8 5x3)

Example Set No.4 Solutions

a) (x + 8)(x + 5) = x2 + 5x + 8x + 40

= x2 + 13x + 40

b) (y + 4) (y 3) = y2 3y + 4y 12

= y2 + y 12

Example Set No. 4 Solution (continued)

c) (5t3 + 4t)(2t2 1) = 10t5 5t3 + 8t3 4t

= 10t5 + 3t3 4t

d) (4 3x)(8 5x3) = 32 20x3 24x + 15x4

= 32 24x 20x3 + 15x4

Multiply (5x3 + x2 + 4x)(x2 + 3x).

Solution 1

5x3 + x2 + 4x

x2 + 3x

15x4 + 3x3 + 12x2

5x5 + x4 + 4x3

5x5 + 16x4 + 7x3 + 12x2

Example No.5

Make sure that you have the terms lined up!

Example No.5 (continued)Multiply (5x3 + x2 + 4x)(x2 + 3x).

Solution 2

4 25 34 331 125 45 x xx x xx Distribute 5x3 Distribute x2 Distribute 4x

5 4 3 25 16 7 12x x x x Combine like terms.

Example No.6

Multiply: (3x2 4)(2x2 3x + 1)

Solution 1

4 3 26 9 3x x x 28 12 4x x

4 3 26 9 11 12 4x x x x

Line up the like terms – leaving a space for the

“missing” x term.

2x2 3x + 1

3x2 4

8x2 + 12x 4 6x4 + 9x3 3x2

6x4 + 9x3 11x2 + 12x 4

Example No.6 Solution