Do Now Given the polynomial 8 -17x x - 20x 4 – Write the polynomial in standard form: _______________________________________ – Classify the polynomial by degree: _______________________________________ – Classify the polynomial by the number of terms: ________________________________ – State the End Behavior: ________________________________ L.T. I can analyze the factored form of a polynomial and write function from its zeros

5.2 Polynomials, Linear Factors, and Zeros Learning Target: I can analyze the factored form of a polynomial and write function from its zeros L.T. I can analyze the factored form of a polynomial and write function from its zeros

Polynomials and Real Roots POLYNIOMIAL EQUIVALENTS 1.Roots 2.Zeros 3.Solutions 4.X-Intercepts 5.Relative Maximum 6.Relative Minimum ROOTS ! Relative Maximum Relative Minimum L.T. I can analyze the factored form of a polynomial and write function from its zeros

The greatest y value of the points in a region is called the local maximum. The least y value among nearby points is called the local minimum. L.T. I can analyze the factored form of a polynomial and write function from its zeros

Linear Factors Just as you can write a number into its prime factors you can write a polynomial into its linear factors. Ex. 6 into 2 & 3 x 2 + 4x – 12 into (x+6)(x-2) L.T. I can analyze the factored form of a polynomial and write function from its zeros

We can also take a polynomial in factored form and rewrite it into standard form. Ex. (x+1)(x+2)(x+3) = foil distribute (x 2 +5x+6)(x+1)=x (x 2 +5x+6)+1 (x 2 +5x+6) = x 3 +6x 2 +11x+6 Standard form L.T. I can analyze the factored form of a polynomial and write function from its zeros

We can also use the GCF (greatest common factor) to factor a poly in standard form into its linear factors. Ex. 2x 3 +10x 2 +12x GCF is 2x so factor it out. We get 2x(x 2 +5x+6) now factor once more to get 2x(x+2)(x+3) Linear Factors L.T. I can analyze the factored form of a polynomial and write function from its zeros

Theorem The expression (x - a) is a linear factor of a polynomial if and only if the value a is a zero (root) of the related polynomial function. If and only if = the theorem goes both ways If (x – a) is a factor of a polynomial, then a is a zero (solution) of the function. and Ifa is a zero (solution) of the function then (x – a) is a factor of a polynomial, L.T. I can analyze the factored form of a polynomial and write function from its zeros

Zeros A zero is a (solution or x-intercept) to a polynomial function. If (x – a) is a factor of a polynomial, then a is a zero (solution) of the function. If a polynomial has a repeated solution, it has a multiple zero. The number of repeats of a zero is called its multiplicity. L.T. I can analyze the factored form of a polynomial and write function from its zeros

A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs. On a graph, a double zero “bounces” off the x axis. A triple zero “flattens out” as it crosses the x axis. L.T. I can analyze the factored form of a polynomial and write function from its zeros

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) Note: two answers are x=2; therefore the multiplicity is 2 Ex: (x-1) 4 (x+3) Note: four answers are x=1; therefore the multiplicity is 4 Ex: y =x(x-1)(x+3) Note: there are no repeat roots, so we say that there is no multiplicity L.T. I can analyze the factored form of a polynomial and write function from its zeros

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 L.T. I can analyze the factored form of a polynomial and write function from its zeros

Write a polynomial given the roots 2, -4, ½ 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 Note that the ½ term becomes (x-1/2). We don’t like fractions, so multiply both terms by 2 to get (2x-1) L.T. I can analyze the factored form of a polynomial and write function from its zeros

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 FACTORED FORM L.T. I can analyze the factored form of a polynomial and write function from its zeros

Let’s Try One Find any multiple zeros of f(x)=x 4 +6x 3 +8x 2 and state the multiplicity L.T. I can analyze the factored form of a polynomial and write function from its zeros

Let’s Try One Find any multiple zeros of f(x)=x 4 +6x 3 +8x 2 and state the multiplicity L.T. I can analyze the factored form of a polynomial and write function from its zeros

Equivalent Statements about Polynomials  -4 is a solution of x 2 +3x-4=0 4 is an x-intercept of the graph of y=x 2 +3x-4 4 is a zero of y=x 2 +3x-4  (x+4) is a factor of x 2 +3x-4 These all say the same thing L.T. I can analyze the factored form of a polynomial and write function from its zeros

Write a polynomial in standard form with zeros at 2, –3, and 0. Polynomials and Linear Factors = (x – 2)(x 2 + 3x)Multiply (x + 3)(x). = x(x 2 + 3x) – 2(x 2 + 3x)Distributive Property = x 3 + 3x 2 – 2x 2 – 6xMultiply. = x 3 + x 2 – 6xSimplify. The function ƒ(x) = x 3 + x 2 – 6x has zeros at 2, –3, and 0. 2–30Zeros ƒ(x) = (x – 2)(x + 3)(x)Write a linear factor for each zero. L.T. I can analyze the factored form of a polynomial and write function from its zeros

Find any multiple zeros of ƒ(x) = x 5 – 6x 4 + 9x 3 and state the multiplicity. Polynomials and Linear Factors ƒ(x) = x 5 – 6x 4 + 9x 3 ƒ(x) = x 3 (x 2 – 6x + 9)Factor out the GCF, x 3. ƒ(x) = x 3 (x – 3)(x – 3)Factor x 2 – 6x + 9. Since you can rewrite x 3 as (x – 0)(x – 0)(x – 0), or (x – 0) 3, the number 0 is a multiple zero of the function, with multiplicity 3. Since you can rewrite (x – 3)(x – 3) as (x – 3) 2, the number 3 is a multiple zero of the function with multiplicity 2. L.T. I can analyze the factored form of a polynomial and write function from its zeros

Graph / Sketch Y = (x-2)(x+9) We know there are zeros at? Can you describe the end behavior? How many turns are there? Sketch it! L.T. I can analyze the factored form of a polynomial and write function from its zeros

Assignment #7 pg odds L.T. I can analyze the factored form of a polynomial and write function from its zeros

Example We can rewrite a polynomial from its zeros. Write a poly with zeros -2, 3, and 3 f(x)= (x+2)(x-3)(x-3) foil = (x+2)(x 2 - 6x + 9) now distribute to get = x 3 - 4x 2 - 3x + 18 this function has zeros at -2,3 and 3 L.T. I can analyze the factored form of a polynomial and write function from its zeros

Finding local Maximums and Minimum Find the local maximum and minimum of x 3 + 3x 2 – 24x Enter equation into calculator Hit 2 nd Trace Choose max or min Choose a left and right bound and tell calculator to guess L.T. I can analyze the factored form of a polynomial and write function from its zeros