Form a polynomial whose real zeros and degree are given. Zeros: -3, 0, 8; degree: 3 Type a polynomial with integer coefficients and a leading coefficient of 1. f(x) = \boxed{} (Simplify your answer.)
The Correct Answer and Explanation is:
To form a polynomial given the real zeros and degree, we use the fact that a polynomial can be factored based on its zeros. The degree of the polynomial tells us how many factors the polynomial will have.
Step 1: Write the factors based on the zeros
The zeros given are: -3, 0, and 8. Each zero corresponds to a factor in the polynomial:
- A zero of −3-3−3 corresponds to the factor (x+3)(x + 3)(x+3).
- A zero of 000 corresponds to the factor xxx.
- A zero of 888 corresponds to the factor (x−8)(x – 8)(x−8).
Step 2: Multiply the factors to form the polynomial
Now, we multiply the factors to get the polynomial:f(x)=(x+3)(x)(x−8)f(x) = (x + 3)(x)(x – 8)f(x)=(x+3)(x)(x−8)
Step 3: Expand the expression
We now expand the factors step by step.
First, multiply (x+3)(x + 3)(x+3) and xxx:(x+3)(x)=x2+3x(x + 3)(x) = x^2 + 3x(x+3)(x)=x2+3x
Now, multiply (x2+3x)(x^2 + 3x)(x2+3x) by (x−8)(x – 8)(x−8):(x2+3x)(x−8)=x2(x−8)+3x(x−8)(x^2 + 3x)(x – 8) = x^2(x – 8) + 3x(x – 8)(x2+3x)(x−8)=x2(x−8)+3x(x−8)
Distribute each term:x2(x−8)=x3−8x2x^2(x – 8) = x^3 – 8x^2×2(x−8)=x3−8x23x(x−8)=3×2−24x3x(x – 8) = 3x^2 – 24x3x(x−8)=3×2−24x
Now, combine all the terms:f(x)=x3−8×2+3×2−24xf(x) = x^3 – 8x^2 + 3x^2 – 24xf(x)=x3−8×2+3×2−24x
Combine like terms:f(x)=x3−5×2−24xf(x) = x^3 – 5x^2 – 24xf(x)=x3−5×2−24x
Step 4: Final result
Thus, the polynomial is:f(x)=x3−5×2−24xf(x) = x^3 – 5x^2 – 24xf(x)=x3−5×2−24x
Explanation:
- The degree of the polynomial is 3, which matches the number of given zeros, and the leading coefficient is 1, as required.
- We started with the factored form using the given zeros and then expanded the expression to obtain the polynomial in standard form.
