1. 6.2 Polynomials and Linear Functions

2. Polynomials and Real Roots
Relative Maximum
POLYNIOMIAL EQUIVALENTS
Roots
Zeros
Solutions
X-Intercepts
Relative Maximum
Relative Minimum
Relative Minimum
ROOTS !

3. Write a polynomial given the roots 0, -3, 3
Put in factored form
y = (x – 0)(x + 3)(x – 3)
y = (x)(x + 3)(x – 3)
y = x(x² – 9)
y = x³ – 9x

4. Write a polynomial given the roots 2, -4, ½
Note that the ½ term becomes (x-1/2). We don’t like fractions, so multiply both terms by 2 to get (2x-1)
Put in factored form
y = (x – 2)(x + 4)(2x – 1)
y = (x² + 4x – 2x – 8)(2x – 1)
y = (x² + 2x – 8)(2x – 1)
y = 2x³ – x² + 4x² – 2x – 16x + 8
y = 2x³ + 3x² – 18x + 8

5. Write the polynomial in factored form. Then find the roots
Write the polynomial in factored form. Then find the roots. Y = 3x³ – 27x² + 24x
Y = 3x³ – 27x² + 24x
Y = 3x(x² – 9x + 8)
Y = 3x(x – 8)(x – 1)
ROOTS?
3x(x – 8)(x – 1) = 0
Roots = 0, 8, 1 8 -8 -1 -9
FACTORED FORM

6. Write the polynomial in factored form. Then find the roots
Write the polynomial in factored form. Then find the roots. Y = x4 + 6x² + 8 8
Y = x4 + 6x² + 8
Y =(x² + 4)(x² + 2)
ROOTS?
(x² + 4)(x² + 2) = 0
Set each group equal to zero and solve
x² + 4 = 0 x² + 2 = 0
x² = -4 x² = -2
X = √-4 X = √-2
X = ±2i x = ±i√2 4 2 6
FACTORED FORM

7. What is Multiplicity?
Multiplicity is when you have multiple roots that are exactly the same. We say that the multiplicity is how many duplicate roots that exist.
Ex: (x-2)(x-2)(x+3)
Ex: (x-1)4 (x+3)
Ex: y =x(x-1)(x+3)
Note: two answers are x=2; therefore the multiplicity is 2
Note: four answers are x=1; therefore the multiplicity is 4
Note: there are no repeat roots, so we say that there is no multiplicity

8. Let’s Try One
Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity

9. Let’s Try One
Find any multiple zeros of f(x)=x4+6x3+8x2 and state the multiplicity