Polynomials P4

Naming Polynomials

• If a does not equal 0, the degree of axn is n.

• Degree of polynomials is the greatest degree of all its terms

• The degree of a nonzero constant is 0.

• The constant 0 has no defined degree.

# Terms Degree

1 – Monomial 1 – Linear

2 – Binomial 2 – Quadratic

3 – Trinomial 3 – Cubic

4+ - Polynomial 4 + - 4th degree, etc.

Practice:3x4 = 4th degree monomial

5xy2= Cubic monomial

3x2 +6x =

Quadratic binomial

3xy +3x +4 = Quadratic Trinomial

3x4 +5xy + 6x + 2= 4th degree polynomial

Cubic Trinomial3x3+6x2+2x =

Definition of a Polynomial in x• A polynomial in x is an algebraic expression of the form

anxn + an-1x

n-1 + an-2xn-2 + … + a1n + a0

where an, an-1, an-2, …, a1 and a0 are real numbers, an ≠ 0,

and n is a non-negative integer.

The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term.

Definition of a Polynomial in x• A polynomial in x is an algebraic expression of the form

anxn + an-1x

n-1 + an-2xn-2 + … + a1n + a0

where an, an-1, an-2, …, a1 and a0 are real numbers, an ≠ 0,

and n is a non-negative integer.

The polynomial is of degree n, an is the leading coefficient, and a0 is the constant term.

Identify the 3x8 + 5x4 + 2…degree?…leading coefficient?…. and constant term?

Standard Form of a Polynomial

Write in order of descending powers of the variable

So…3x + 5x8 - 9x3 + 10should be written

5x8 - 9x3 +3x +10

Adding and Subtracting Polynomials (Ex#1)

Perform the indicated operations and simplify:(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)

Solution(-9x3 + 7x2 – 5x + 3) + (13x3 + 2x2 – 8x – 6)= (-9x3 + 13x3) + (7x2 + 2x2) + (-5x – 8x) + (3 – 6) Group like terms.

= 4x3 + 9x2 – (-13x) + (-3) Combine like terms.

= 4x3 + 9x2 + 13x – 3

The product of two monomials is obtained by using properties of exponents. For example,

(-8x6)(5x3) = -8·5x6+3 = -40x9

Multiply coefficients and add exponents.

Furthermore, we can use the distributive property to multiply a monomial and a polynomial that is not a monomial. For example,

3x4(2x3 – 7x + 3) = 3x4 · 2x3 – 3x4 · 7x + 3x4 · 3 = 6x7 – 21x5 + 9x4.

monomial trinomial

Multiplying Polynomials (Ex #2)

Multiplying Polynomials when Neither is a Monomial (Ex #3)• Multiply each term of one polynomial by each term of the

other polynomial. Then combine like terms.

Using the FOIL Method to Multiply Binomials

(ax + b)(cx + d) = ax · cx + ax · d + b · cx + b · d Product of

First termsProduct of

Outside termsProduct of

Inside termsProduct ofLast terms

firstlast

inner

outer

Ex #3

Multiply: (3x + 4)(5x – 3).

Text Example

Multiply: (3x + 4)(5x – 3).

Solution 

(3x + 4)(5x – 3) = 3x·5x + 3x(-3) + 4(5x) + 4(-3)= 15x2 – 9x + 20x – 12= 15x2 + 11x – 12 Combine like terms.

firstlast

inner

outer

F O I L

The Product of the Sum and Difference of Two Terms (ex #4)

DIFFERENCE OF SQUARES

• The product of the sum and the difference of the same two terms is the square of the first term minus the square of the second term.

(A B)(A B) A2 B2

The Square of a Binomial Sum (Ex #5)

PERFECT SQUARE TRINOMIAL

• The square of a binomial sum is first term squared plus 2 times the product of the terms plus last term squared.

(A B)2 A2 2AB B2

The Square of a Binomial Difference

• The square of a binomial difference is first term squared minus 2 times the product of the terms plus last term squared.

(A B)2 A2 2AB B2

Let A and B represent real numbers, variables, or algebraic expressions. 

Special Product ExampleSum and Difference of Two Terms(A + B)(A – B) = A2 – B2 (2x + 3)(2x – 3) = (2x) 2 – 32

= 4x2 – 9

Squaring a Binomial(A + B)2 = A2 + 2AB + B2 (y + 5) 2 = y2 + 2·y·5 + 52

= y2 + 10y + 25(A – B)2 = A2 – 2AB + B2 (3x – 4) 2 = (3x)2 – 2·3x·4 + 42

= 9x2 – 24x + 16

Cubing a Binomial(A + B)3 = A3 + 3A2B + 3AB2 + B3 (x + 4)3 = x3 + 3·x2·4 + 3·x·42 + 43

= x3 + 12x2 + 48x + 64(A – B)3 = A3 – 3A2B – 3AB2 + B3 (x – 2)3 = x3 – 3·x2·2 – 3·x·22 + 23

= x3 – 6x2 – 12x + 8

Special Products

Text Example

Multiply: a. (x + 4y)(3x – 5y) b. (5x + 3y) 2

SolutionWe will perform the multiplication in part (a) using the FOIL method. We will multiply in part (b) using the formula for the square of a binomial, (A + B) 2. a. (x + 4y)(3x – 5y) Multiply these binomials using the FOIL method.

= (x)(3x) + (x)(-5y) + (4y)(3x) + (4y)(-5y) = 3x2 – 5xy + 12xy – 20y2

= 3x2 + 7xy – 20y2 Combine like terms.

• (5 x + 3y) 2 = (5 x) 2 + 2(5 x)(3y) + (3y) 2 (A + B) 2 = A2 + 2AB + B2

= 25x2 + 30xy + 9y2

F O I L

Example

• Multiply: (3x + 4)2.

( 3x + 4 )2 =(3x)2 + (2)(3x) (4) + 42 =9x2 + 24x + 16

Solution:

Polynomials