1. Polynomials P4

2. Naming Polynomials Practice: 3x4 = 4th degree monomial 5xy2=
# Terms
Degree
1 – Monomial
1 – Linear
2 – Binomial
2 – Quadratic
3 – Trinomial
3 – Cubic
4+ - Polynomial
th degree, etc.
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.
Practice:
3x4 =
4th degree monomial
5xy2=
Cubic monomial
3xy +3x +4 =
Quadratic Trinomial
3x4 +5xy + 6x + 2=
4th degree polynomial
3x2 +6x =
Quadratic binomial
3x3+6x2+2x =
Cubic Trinomial

3. 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.

4. 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.
Identify the 3x8 + 5x4 + 2
…degree?
…leading coefficient?
…. and constant term?

5. Standard Form of a Polynomial
Write in order of descending powers of the variable So…3x + 5x8 - 9x should be written
5x8 - 9x3 +3x +10

6. 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

7. Multiplying Polynomials (Ex #2)
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

8. 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.

9. 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

10. Ex #3
Multiply: (3x + 4)(5x – 3).

11. 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

12. 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.

13. 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.

14. 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.

15. Special Products
Let A and B represent real numbers, variables, or algebraic expressions. 
Special Product Example
Sum and Difference of Two Terms
(A + B)(A – B) = A2 – B (2x + 3)(2x – 3) = (2x) 2 – 32
= 4x2 – 9
Squaring a Binomial
(A + B)2 = A2 + 2AB + B (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 + B (x + 4)3 = x3 + 3·x2·4 + 3·x·
= x3 + 12x2 + 48x + 64
(A – B)3 = A3 – 3A2B – 3AB2 + B (x – 2)3 = x3 – 3·x2·2 – 3·x·
= x3 – 6x2 – 12x + 8

16. 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 – 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

17. Example ( 3x + 4 )2 =(3x)2 + (2)(3x) (4) + 42 =9x2 + 24x + 16
Multiply: (3x + 4)2.
Solution:
( 3x + 4 )2 =(3x)2 + (2)(3x) (4) =9x2 + 24x + 16

18. Polynomials