6.2 Polynomials and Linear Functions

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

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

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

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

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

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

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

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