Polynomial Graphs: Zeroes and Multiplicity Unit 5 Standard:MM3A1

Zeroes Also known as Where the graph crosses the x-axis x-intercept Solution Roots Where the graph crosses the x-axis

f(x)=x2 + 2x What are the zeroes? How many zeroes? Degree? {-2, 0} 2 2

g(x) = -2x2 + x What are the zeroes? How many zeroes? Degree? {0, 1/2}

h(x)= x3 - x What are the zeroes? How many zeroes? Degree? {-1, 0, 1}

j(x) = -x3 +2x2 +3x What are the zeroes? How many zeroes? Degree? {-1, 0, 3} 3 3

k(x)= x4 -5x2 + 4 What are the zeroes? How many zeroes? Degree? {-2,-1, 1, 2} 4 4

l(x) = -(x4 -5x2 + 4) What are the zeroes? How many zeroes? Degree? {-2,-1, 1, 2} 4 4

m(x)=1/2(x5+x4-7x3 -22x2 + 24x) What are the zeroes? How many zeroes? Degree? {-4,-3, 0, 1, 2} 5 5

n(x)=-1/2(x5+x4-7x3 -22x2 + 24x) What are the zeroes? How many zeroes? Degree? {-4, -3, 0, 1, 2} 5 5

What is the relationship between degree and number of zeroes? They are the same Graph the following polynomial: y=(x-1)(x-3) x-intercepts? Zeroes? Number of zeroes? Degree? {1, 3} 1, 3 2 2 (x-1)(x-3)=x2–x-3x+3 = x2-4x+3 Zeroes and degree

(x-1)(x-3)=x2–x-3x+3 = x2-4x+3 Standard form Factored form To find the zeroes, set each factor =0 (x-1)=0 X=1 (x-3)=0 X=3

Write the standard form of the equation for the polynomial function with the given zeroes Example: Steps: 0, 4 and -2 (x-0)(x-4)(x+2) f(x) = x(x-4)(x+2) f(x) = x(x2 +2x-4x-8) f(x)= x(x2 - 2x - 8) f(x) = x3 - 2x2 – 8x Use the zeroes to write the factors: (x - the zero) Write as a function in factored form. Multiply (Foil) and simplify

Write the standard form of the equation for the polynomial function with the given zeroes 1, 3(multiplicity 2) f(x) = (x-1)(x-3)(x-3) f(x) = (x2 -3x – x +3)(x-3) f(x) = (x2 - 4x + 3)(x–3) f(x) = x3 - 3x2 - 4x2 + 12x +3x– 9 f(x) = x3 - 7x2+15x-9 Multiplicity means that a zero is used as a factor more than once

For each function, determine the zeroes and their multiplicity y=(x-3)(x+2)2 x-3=0 x=3 x+2=0 x=-2 with multiplicity 2 3, -2(mult. 2) For each function, determine the zeroes and their multiplicity

For each function, determine the zeroes and their multiplicity 2. y=x(x-5)10 (x+4)2 x=0 x-5=0 x=5 with multiplicity 10 x+4=0 x=-4 with multiplicity 2 0, 5 (mult 10), -4(mult. 2) For each function, determine the zeroes and their multiplicity