Form a polynomial whose zeros and degree are given.
​Zeros: minus4​, 4​, 5​;  ​ degree: 3
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Part 1
Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below.
​f(x)equals

enter your response here ​(Simplify your​ answer.)

The zeros of the polynomial are given as \( -4 \), \( 4 \), and \( 5 \), and the degree of the polynomial is 3. To form a polynomial, we use the fact that if \( c \) is a zero of a polynomial, then \( (x - c) \) is a factor.

So, for the given zeros, the polynomial can be written as:

\[
f(x) = (x - (-4))(x - 4)(x - 5)
\]

Simplify the factors:

\[
f(x) = (x + 4)(x - 4)(x - 5)
\]

Now, expand the factors. First, use the difference of squares on \( (x + 4)(x - 4) \):

\[
f(x) = (x^2 - 16)(x - 5)
\]

Next, distribute \( (x - 5) \) over \( (x^2 - 16) \):

\[
f(x) = x^2(x - 5) - 16(x - 5)
\]

\[
f(x) = x^3 - 5x^2 - 16x + 80
\]

Thus, the polynomial is:

\[
f(x) = x^3 - 5x^2 - 16x + 80
\]

This is a polynomial with integer coefficients and a leading coefficient of 1.

Would you like more details or explanations on any of the steps?

Here are 5 related questions you might find interesting:
1. How do you find the degree of a polynomial given its expression?
2. What is the difference of squares, and how does it apply in polynomial expansion?
3. How can you verify if a polynomial has certain zeros?
4. How do the coefficients of a polynomial relate to its graph's shape?
5. What is synthetic division and how does it help with polynomial roots?

**Tip:** Always check your work by substituting the zeros back into the polynomial to confirm they yield 0!

Form a polynomial whose zeros and degree are given. Zeros: -4, 4, 5; degree: 3. Type a polynomial with integer coefficients and a leading coefficient of 1.

Learn how to form a cubic polynomial with integer coefficients, given its zeros (-4, 4, and 5) and degree 3. This solution includes factoring, using the difference of squares, and expanding polynomials. Suitable for high school students (Grades 9-12).

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