Unit 5 polynomial functions homework 3 zeros and multiplicity answers

Let's call the representation of the polynomial function ƒ. We know that ƒ has zeroes at 5/2 (multiplicity 2), 3 (multiplicity 1), and 0 (multiplicity 4). We can use this to express factors of ƒ:

x = 5/2,

(x - 5/2) = 0 (subtract 5/2 from both sides)

2x - 5 = 0 (multiply by 2 on both sides)

x = 3

(x - 3) = 0 (subtract 3 from both sides)

and

(x) = 0 (no need to rearrange)

Now express ƒ as the product of the factors using exponents for multiplicity:

ƒ(x) = (2x-5)2(x-3)1(x)4,

This is a polynomial function with the specified zeros. Now we only need to express this in standard form, which is of the form: ax7 + bx6 + cx5 + dx4 + ex3 + fx2 + gx1 + hx0. (We now it will be degree 7 because of the number of zeros including multiplicity (2 + 1 + 4 = 7))

So now we expand the polynomial to standard form by multiplying the factors together:

(2x - 5)2 = (2x - 5)(2x - 5)

= 4x2 - 20x + 25

(4x2 - 20x + 25)(x-3) = 4x3 - 20x2 + 25x - 12x2 + 60x - 75

= 4x3 - 32x2 + 85x -75

(4x3 - 32x2 + 85x -75)x4 = 4x7 - 32x6 + 85x5 -75x4

So our answer is:

ƒ(x) = 4x7 - 32x6 + 85x5 -75x4

Donovan B. answered • 12/09/19

Your mind can achieve great things!

ANSWER:

f(x)=x7-8x6+85x5/4-75x4/4

EXPLANATION:

Write the question out as a polynomial function

f(x)=(x-5/2)2(x-3)1(x-0)4

Simplify the equation

f(x)=(x-5/2)(x-5/2)(x-3)(x4)

Multiply the equation to write in standard form

(Values in decreasing order)

f(x)=x7-8x6+85x5/4-75x4/4

HOPE THIS HELPS :)

Arturo O. answered • 12/09/19

Experienced Physics Teacher for Physics Tutoring

Set this up as a product of factors containing the zeros, with the multiplicity as a power. You have

zero 5/2, multiplicity 2

zero 3, multiplicity 1

zero 0, multiplicity 4

f(x) = A(x - 5/2)2 (x - 3)1(x - 0)4

where the leading coefficient is A, and A is real and ≠ 0. There is not sufficient information in the problem statement to find A. Anyway, expand and simplify f(x) to get it in standard form. You will get a polynomial of degree 7 in the form

f(x) = a7x7 + a6x6 + .... + a1x + a0