Polynomials
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Polynomials. The Degree of ax n. If a does not equal 0, the degree of ax n is n . The degree of a nonzero constant is 0. The constant 0 has no defined degree. Definition of a Polynomial in x. A polynomial in x is an algebraic expression of the form

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Polynomials

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Polynomials


The Degree of axn

  • If a does not equal 0, the degree of axn is n. The degree of a nonzero constant is 0. The constant 0 has no defined degree.


Definition of a Polynomial in x

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

  • anxn + an-1xn-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.


Text Example

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


Multiplying Polynomials

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 when Neither is a Monomial

  • Multiply each term of one polynomial by each term of the other polynomial. Then combine like terms.


Using the FOIL Method to Multiply Binomials

last

first

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

Product of

First terms

Product of

Outside terms

Product of

Inside terms

Product of

Last terms

inner

outer


Text Example

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 – 12Combine like terms.

last

first

F

O

I

L

inner

outer


The Product of the Sum and Difference of Two Terms

  • 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.


The Square of a Binomial Sum

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


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.


Special Products

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

Special ProductExample

Sum 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


Text Example

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

  • Solution

  • We 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 – 20y2Combine 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.

Solution:

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


Polynomials


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