1. Polynomial Functions
IM3
Ms.Peralta

2. What you’ll learn today:
Polynomial functions.
Power of functions: examples.
Add and subtract polynomials.
Multiply polynomials.

3. f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao
Polynomials
An expression in the form of
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao
where n is a non-negative integer and a2, a1, and a0 are real numbers.
The function is called a polynomial function of x with degree n.
A polynomial is a monomial or a sum of terms that are monomials.
Polynomials can NEVER have a negative exponent or a variable in the denominator!
The term containing the highest power of x is called the leading coefficient, and the power of x contained in the leading terms is called the degree of the polynomial.

4. Examples of Polynomials
Degree
Name
Example
Constant 5 1
Linear
3x+2 2
Quadratic
X2 – 4 3
Cubic
X3 + 3x + 1 4
Quartic
-3x4 + 4
Quintic
X5 + 5x4 - 7

5. polynomial function; f (x) = –2x + 13; degree 1; type: linear;
Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient.
GUIDED PRACTICE
1. f (x) = 13 – 2x
polynomial function; f (x) = –2x + 13;
degree 1; type: linear;
leading coefficient: –2
2. p (x) = 9x4 – 5x –2 + 4
not a polynomial function
polynomial function;
h(x) = 6x2 – 3x + π ; degree 2,
type: quadratic;
leading coefficient: 6
3. h (x) = 6x2 + π – 3x

6. Add & Subtract Polynomials
Monomial: 1 term
These are all polynomials
Binomial: 2 terms
Trinomial: 3 terms
Adding Polynomials: Combine the like terms
Like Terms – Terms that have the same variables with the same exponents on them
Combining Like Terms: Add the coefficients of each all like terms. Ex. 3x + (-5x) = [3 + (-5)]x = -2x

7. Example 1:
Rewrite
Combine Like Terms

8. Example 2: Add 3y3 – 2y2 – 7y and –4y2 + 2y – 5
(3y3 – 2y2 – 7y) + (–4y2 + 2y – 5)
3y3 – 2y2 – 4y2 – 7y + 2y – 5
Gather like terms
3y3 – 6y2 – 5y – 5
Combine like terms

9. Example 3: Subtract 5z2 – z + 3 from 4z2 + 9z – 12
(4z2 + 9z – 12) – (5z2 – z + 3)
Remember to distribute the – through the ( )
4z2 + 9z – 12 – 5z2 + z – 3
4z2 – 5z2 + 9z + z – 12 – 3
Gather like terms
–z2 + 10z – 15
Combine like terms

10. You try it! Find the sum 1. (t2 – 6t + 2) + (5t2 – t – 8)
GUIDED PRACTICE
Find the sum
1. (t2 – 6t + 2) + (5t2 – t – 8)
t2 + 5t2 – 6t – t + 2 – 8
6t2 – 7t – 6
Find the difference
2. (8d – 3 + 9d3) – (d3 – 13d2 – 4)
8d – 3 + 9d3 – d3 + 13d2 + 4
9d3 – d3 + 13d2 + 8d – 3 + 4
8d3 + 13d2 + 8d + 1

11. There are three techniques you can use for multiplying polynomials.
It’s all about how you write it…
Distributive Property
FOIL
Box Method

12. Remember, FOIL reminds you to multiply the:
First terms
Outer terms
Inner terms
Last terms

13. The FOIL method is ONLY used when you multiply 2 binomials
The FOIL method is ONLY used when you multiply 2 binomials. It is an acronym and tells you which terms to multiply. Use the FOIL method to multiply the following binomials: (y + 3)(y + 7).

14. (y + 3)(y + 7) F tells you to multiply the FIRST terms of each binomial. O tells you to multiply the OUTER terms of each binomial. I tells you to multiply the INNER terms of each binomial. L tells you to multiply the LAST terms of each binomial.
y2 + 7y + 3y + 21
Combine like terms.
y2 + 10y + 21

15. Multiply (2x - 5)(x2 - 5x + 4)
You cannot use FOIL because they are not BOTH binomials. You must use the distributive property.
2x(x2 - 5x + 4) - 5(x2 - 5x + 4)
2x3 - 10x2 + 8x - 5x2 + 25x - 20
Group and combine like terms.
2x3 - 10x2 - 5x2 + 8x + 25x - 20
2x3 - 15x2 + 33x - 20

16. Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH binomials. You must use the distributive property or box method. x2
-5x +4 2x -5
2x3
-10x2
+8x
Almost done!
Go to the next slide!
-5x2
+25x
-20

17. Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!+4 2x -5
2x3
-10x2
+8x
-5x2
+25x
-20
2x3 – 15x2 + 33x - 20