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

Polynomials


The degree of ax n

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

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

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

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

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

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 example1

Text Example

Multiply: (3x + 4)(5x – 3).


Text example2

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 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

  • 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

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


Special products

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 example3

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

Example

  • Multiply: (3x + 4)2.

Solution:

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


Polynomials1

Polynomials


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