Download 5.1 Notes: Polynomial Functions monomial: a real number, variable

5.1 Notes: Polynomial Functions
monomial: a real number, variable, or product of one real number and one or more variables
degree of monomial: the exponent of the variable(s)
polynomial: a monomial or sum of monomials
degree of polynomial: the greatest degree among its monomial terms
*Constant Difference: From the y-values on a table, the differences between each term is constant (this tells you the
degree)
polynomial function: the degree of the polynomial function
standard form: arranges the terms by degree in descending numerical order
Leading Term: axn
Up and Up
a Positive
n Even
Down and Up
a Positive
n Odd
Down and Down
a Negative
n Even
Examples: (leave 2 lines)—can fit 2 or 3 problems on each line
Write in standard form. Classify by degree and number of terms.
1) 7x + 3x + 5
2) 2m2 – 3+7m
3) x2 – x4 +2x2
Determine the end behavior of the graph of each polynomial function.
4) y=-3x + 6x2–1
5) y=-3 – 6x5–9x8
6) y=-x3–x2+3
Determine the degree of the polynomial:
7)
x
y
-2
-15
-1
-9
0
-9
1
-9
2
-3
Up and Down
a Negative
n Odd
5.2 Polynomials, Linear, Factors, and Zeros
Factoring:
-Find GCF
-Use Berry Method/unFOIL
Zero Product Property:
-Set all equal to zero
-Solve for roots
-Graph roots
-Use end behavior rules to sketch function
Standard Form Given Zeros:
-Change signs (Example: -2  (x+2))
-Multiply using FOIL and exponent rules
-Write in standard form
Examples: (Leave 1 line for 1-3, 4-5 lines for 4 & 5, and about 3 lines for 6 & 7)
Write in factored form. Check by multiplication.
1) x3+7x2+10x
2) x3-81x
3) x3+8x2+16x
Find the zeros of each function. Then graph the function.
4) y=(x-1)(x+2)
5) x(x+5)(x-8)
Write a polynomial in standard form with the given zeros.
6) x=0, 2, 4
1
7) x=2, -1, 5, 3
5.3 Solving Polynomial Equations
6x3+15x2+9x
8x2+10x-3
4x2+12x+9
x2-36
2x3+2x2+3x+3
x3-27
Examples: Leave 2-3+ lines for each problem
Find the real or imaginary solutions of each equation by factoring.
1)
2)
3)
4)
5)
6)
7)
8)
2x3 – 5x2 = 3x
3x4 + 12x2 = 6x3
(x2 – 1)(x2 + 4) = 0
x5 + 4x3 = 5x4 – 2x3
x4 = 16
x3 = 8x – 2x3
x3 = 1
x(x2 + 8) = 8(x + 1)
DO THESE EXAMPLES!
BEFORE CLASS!
5.4 Notes: Dividing Polynomials
Long Division:
-Set up Problem with binomial on outside & polynomial on inside
-Divide, multiply, subtract like usual
*May get a remainder
*If no remainder, that binomial is a factor of the polynomial
Synthetic Division:
-Solve for x of binomial (normally just change sign)
-Write coefficients of polynomial in descending order (0s for placeholders)
-Bring down first term
-Multiply answer by divisor, then rewrite & add to next coefficient
-Repeat; last number is remainder (if 0, then binomial is a factor)
Remainder Theorem: P(x) is a polynomial, then P(a) is the remainder.
*Plug in and evaluate function to check remainder*
Examples: (Leave 5-6 lines for long division, 3 lines for synthetic)
Divide using long division:
1) (x2– 3x – 40) ÷ (x+5)
2) (x3+ 3x2 –x +2) ÷ (x-1)
Divide using synthetic division:
3) (2x2+7x+11) ÷ (x+2)
4) (x3+5x2+11x + 15) ÷ (x+3)
5) (6x2+7x+2) ÷ (2x+1)
Use synthetic division and the Remainder Theorem to find P(a).
6) P(x) = x3+4x2+4x; a= -2
7) P(x) = x3+7x2+4x; a= -2
5.5 Notes: Theorems About Roots of Polynomial Equations
Rational Root Theorem:
factor of constant term
factor of leading coefficient
*To find actual roots,
in root
for x value;
if it by
=0,lead
thencoefficient’s
it’s a root) factors)
± use Remainder Theorem (plug
*(divide
constant’s
factors
1) Use “Table” button on calculator
2) Type Equation & press enter
3) Set Start=0, Step=1, and Ask-x; Press OK
4) Type in all x values (it will do the work for you) *MUST PUT TABLE IN WORK*
5) All y values that =0 give you your x answers
Conjugate Root Theorem:
Rational Coefficients: If a + √𝑥 is a root, then a−√𝑥 is a root also
Real Coefficients: If a + bi is a root, then a−bi is a root also
*If given roots: put into factored form and multiply to get function.
**remember to use both i AND –i if only one is given**
Examples: (Leave 3+ lines for each)
Use the Rational Root Theorem to list all possible roots for each equation. Then find any actual roots.
1) x3 – 4x + 1 = 0
2) 2x3 – 5x + 4 =0
3) 2x3 + x2 – 7x – 6=0
Write a polynomial function with rational coefficients so that P(x)=0 has the given roots.
4) 5 and 9
5) -4 and 2i
5.6 The Fundamental Theorem of Algebra

The degree of a polynomial tells you how many roots it has. (Ex: x2=2 roots, x5=5 roots, etc.)
**These roots could be real or imaginary and/or have a multiplicity greater than 1**

How to find all of the roots without a calculator:
1. Try to factor (Grouping, GCF, etc.)
2a. If you can factor, factor all the way and set each factor =0
2b. If you can NOT factor, write all possible roots (± constant factors ÷ lead coefficient factors)
3. Try each root by using the remainder theorem (plug in for x) until you find a remainder of 0.
4. Use this factor in synthetic division (*do not change sign again; you already did)
5. Record root you used as x=__
6. Factor your new polynomial after dividing
7. Use step 2a to finish problem and add to answer from step 5
Examples: Leave 1 line for problems 1-3 total & Leave 5+ entire lines for each problem 4-7 (do not break into columns)
Find the number of roots:
1) x3+2x2-5=0
2) x5-17=0
3) x14-x7+x3-2=0
Without using a calculator, find all the roots of each equation.
4)
5)
6)
7)
y=x3-4x2+9x-36
x5-x4-3x3+3x2-4x+4=0
y=x4-x3-5x2-x-6
x5-3x3-4x=0
5.7 The Binomial Theorem
Expand: multiplying an expression (normally using the binomial theorem)
Pascal’s Triangle: a formula for expanding a binomial:
Row
Power
0
(a+b)0
1
(a+b)1
2
(a+b)2
3
(a+b)3
Expanded Form
1
1a1+1b1
1a2+2a1b1+1b2
1a3+3 a2b1+3a1b2+1b3
…etc…
 Coefficients of expanded form match Pascal’s Triangle
*exponents always add to power you are expanding to
Examples: (Leave 2 lines each)
1) (x+3)4
2) (3-z)3
3) (x+y)5
5.9 Transforming Polynomial Functions
y=a(x-h) + k
a=stretch (a>1) or compression (a<1)
h=horizontal translation *change sign
k=vertical translation
*put – sign in front of () if reflected
Examples: (Leave 1-2 lines each)
Determine the cubic function that is formed from the parent function y=x3.
1)
Vertical stretch by the factor 3
Reflection in the x-axis
Vertical translation 4 units up
2)
Vertical compression by factor 1/2
Reflection in the x-axis
Horizontal translation 3 units to the right
Vertical translation 2 units up
3)
Vertical stretch by the factor 2
Horizontal translation 3 units left
Vertical translation 4 units down