Question

Write a polynomial function f of least degree that has a leading coefficient of 1 and the given zeros -4,-2,5. write the function in standard form.

Answer 1

To create a polynomial with zeros -4, -2, 5, and a leading coefficient of 1, multiply the factors (x + 4), (x + 2), and (x - 5) together and expand to the standard form f(x) = x^3 + 6x^2 - 28x - 40.

Step-by-step explanation:

To write a polynomial function f of least degree with the given zeros -4, -2, 5 and a leading coefficient of 1, we first write the factors associated with each zero. The polynomial will be the product of these factors.

The factor for the zero -4 is (x - (-4)) or (x + 4), the factor for the zero -2 is (x - (-2)) or (x + 2), and the factor for the zero 5 is (x - 5). Multiplying these factors together gives us the polynomial in factorized form:

f(x) = (x + 4)(x + 2)(x - 5).

Next, we need to expand these factors to write the polynomial in standard form:

f(x) = x^3 + (4+2)x^2 - (20+8)x - 40.

Combining like terms results in the standard form of the polynomial:

f(x) = x^3 + 6x^2 - 28x - 40.

This is the required polynomial function in standard form with the least degree that satisfies the given conditions.

Answer 2

Step-by-step explanation:

This is a third degree polynomial since we have 3 zeros. We find these zeros by factoring the given polynomial. The zeros of a polynomial are where the graph of the function goes through the x-axis (where y = 0). If x = -4, the factor that gives us this value is (x + 4) = 0 and solving that for x, we get x = -4. If x = -2, the factor that gives us that value is (x + 2) = 0 and solving that for x, we get x = -2. Same for the 5. The way we find the polynomial that gave us these zeros is to go backwards from the factors and FOIL them out. That means that we need to find the product of

(x + 4)(x + 2)(x - 5). Do the first 2 terms, then multiply in the third.

`(x+4)(x+2)=x^2+2x+4x+8`, which simplifies to

`x^2+6x+8`

No we multiply in the final factor of (x - 5):

`(x^2+6x+8)(x-5)=x^3+6x^2+8x-5x^2-30x-40` which simplifies to

`f(x)=x^3+x^2-22x-40`

If you are aware of the method for factoring higher degree polymomials, which is to use the Rational Root Theorem and synthetic division, you will see that this factors to x = -4, -2, 5. If you know how to use your calculator, you will find the same zeros in your solving polynomials function in your apps.