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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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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 – 12 Combine 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

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example
Example
  • Multiply: (3x + 4)2.

Solution:

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

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