Polynomial Functions - Zeroes and Multiplicity

Question

A polynomial function has zeros at 5/2 (multiplicity 2), 3 (multiplicity 1), and 0 (multiplicity 4). Write a function in standard form that could represent this function.

I'm confused as to how to work this out, any help would be greatly appreciated.

Answer 1

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

Answer 2

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 :)

Answer 3

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