Answer:
To write a polynomial function in standard form that has a zero at 4 with a multiplicity of 3 and a zero at 0 with a multiplicity of 1, we can use the concept of polynomial factorization.
Given:
Zero at 4 with multiplicity 3
Zero at 0 with multiplicity 1
To represent this function, we can start with the linear factors:
(x - 4)(x - 4)(x - 4)(x - 0)
Simplifying the factors, we have:
(x - 4)^3(x - 0)
Expanding further:
(x - 4)(x - 4)(x - 4)x
Now, let's multiply the factors to obtain the polynomial in standard form:
(x - 4)(x - 4)(x - 4)x = (x^3 - 12x^2 + 48x - 64)x
Multiplying, we get:
x^4 - 12x^3 + 48x^2 - 64x
Therefore, the polynomial function in standard form that represents the given conditions is:
f(x) = x^4 - 12x^3 + 48x^2 - 64x.
Explanation:
To write a polynomial function in standard form that has a zero at 4 with a multiplicity of 3 and a zero at 0 with a multiplicity of 1, we can use the zero-product property and the fact that if a zero has a multiplicity greater than 1, the corresponding linear factor appears multiple times.
Given:
Zero at 4 with multiplicity 3
Zero at 0 with multiplicity 1
To represent this function, we can start with the linear factors:
(x - 4)(x - 4)(x - 4)(x - 0)
Expanding this expression, we have:
(x - 4)^3(x - 0)
Now, let's simplify and rewrite it in standard form:
(x - 4)^3(x)
Expanding further:
(x - 4)(x - 4)(x - 4)x
Finally, let's multiply the factors to obtain the polynomial in standard form:
(x^3 - 12x^2 + 48x - 64)x
Multiplying, we get:
x^4 - 12x^3 + 48x^2 - 64x
Therefore, the polynomial function in standard form that represents the given conditions is:
f(x) = x^4 - 12x^3 + 48x^2 - 64x.