Find a polynomial function of degree 4 with −2 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1.
f(x)=
Thank you!
"with −2 as a zero of multiplicity 3" ---> (x+2)^3
"0 as a zero of multiplicity 1. " ---- x
f(x) = a x(x+2)^3 , where a is a constant , a≠0
To find a polynomial function of degree 4 with −2 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1, we can use the zero product property.
The zero product property states that if a polynomial function has a zero 𝑟 of multiplicity 𝑚, then the polynomial can be factored as (𝑥 − 𝑟) raised to the power of 𝑚.
In this case, we have:
For the zero -2 with multiplicity 3, the factor is (𝑥 - (-2))^3 = (𝑥 + 2)^3
For the zero 0 with multiplicity 1, the factor is (𝑥 - 0)^1 = 𝑥
To construct the polynomial function, we multiply all the factors together:
𝑓(𝑥) = (𝑥 + 2)^3 * 𝑥
Expanding this multiplication, we get:
𝑓(𝑥) = (𝑥 + 2)(𝑥 + 2)(𝑥 + 2)(𝑥)
Simplifying further, we can use the distributive property and multiply the factors:
𝑓(𝑥) = (𝑥 + 2)(𝑥^2 + 4𝑥 + 4)(𝑥)
Finally, we can multiply the terms using the distributive property again:
𝑓(𝑥) = (𝑥^3 + 4𝑥^2 + 4𝑥)(𝑥)
𝑓(𝑥) = 𝑥^4 + 4𝑥^3 + 4𝑥^2 + 𝑥^2 + 4𝑥 + 4𝑥
𝑓(𝑥) = 𝑥^4 + 4𝑥^3 + 5𝑥^2 + 8𝑥
Therefore, the polynomial function of degree 4 with −2 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1 is:
𝑓(𝑥) = 𝑥^4 + 4𝑥^3 + 5𝑥^2 + 8𝑥