1. What is a monomial
In one variable; is the product of a constant and a variable raised to a non negative integer power. The form is axk
a is the constant
x is the variable
k is the degree but must be ≥ 0 and an integer.

2. Polynomial Functions Polynomial is a monomial or a sum of monomials
Some terms to know:
Polynomial is a monomial or a sum of monomials
Polynomial Function: function in the form f(x) = anxn + an-1xn-1 + … a0.
an is the leading coefficient
n is the degree,
and a0 is the constant.
What is the degree, leading coefficient, and constant:
5x3 + 4x – 3
Degree is 3, leading coefficient is 5 and constant is -3

3. Simplifying Polynomials
Adding and subtracting polynomials: Only add or subtract the coefficients of terms with the same base and same exponent.
(3y2 -6y4 + 5 – 6y) + (5y4 – 6y2 + 4y)
-y4 – 3y2 – 2y + 5
(8y4 -2y2 + y – 4) – (3y3 – 12y2 + 8y)
-3y3 + 12y2 – 8y
8y4 – 3y3 +10y2 -7y – 4

4. Practice adding polynomials:
(x3 + 3x2 + 2) + (x2 – 4x + 4)
Practice subtracting polynomials:
(x2 – 3x – 4) – (x3 – 3x2 + x + 5)

5. Multiplying Polynomials
Distribute each term to each term in the other parenthesis then combine like terms.
(x + 4)3
Cube the first term; square the first term and multiply by the second term and 3; Square the second term multiply by the first term and 3; Cube the second term.
(5c2 – 4)(2c2 + c – 3)
10c4 + 5c3 – 15c2
-8c2 -4c +12
x3 + 3((x)2 4) + 3(x(4)2) + 43
x3 + 12x x + 64
10c4 + 5c3 – 23c2 – 4c + 12

6. More examples (x – 3)(x – 3) x2 – 6x + 9 5c2 + 7c + 1 -2c2 + 6c – 8
* You can’t just square the x and the 3 you will miss the middle term
*remember distribute the negative to each term.

7. Practice Use multiplication property to simplify the polynomial:
(3x + 1) (2x + 1)
Use the rule for exponents to simplify the polynomial:
(x + 1)3

8. Practice
Use multiplication and subtraction property to simplify the polynomial:
8(4x3 – 3x2 – 1) – 6(4x3 + 8x – 2)