Form a polynomial whose real zeros and degree are given. Zeros: -5,-2, 4, 5, degree: 4 Type a polynomial with integer coefficients and a leading coefficient of 1.

Question

I am looking for a full breakdown on how to solve this and what correct formulas to use.

Answer 1

To find a polynomial with zeros of -5, -2, 4, and 5 and a degree of 4, you must first check to see if there are any non-real zeros. Since 4 real zeros are given and the degree of the polynomial is 4, you can conclude that there are no non-real zeros. Additionally, since the leading coefficient is stated to be 1, it can be concluded that each term contains an x rather than any multiple of x (such as 2x, 3x etc.). The next step is to write out the factored form of the polynomial with the zeros given like so:

(x+5)(x+2)(x-4)(x-5)

After this step, I personally like to multiple out the left two terms and right two terms first using the foil method:

(x² + 2x + 5x + 10)(x² - 5x - 4x +20)

After simplifying, you get:

(x² + 7x + 10)(x² - 9x +20)

Then, you multiply out all of the terms again using a longer foil method:

(x⁴ - 9x³ + 20x² + 7x³ - 63x² + 140x + 10x² - 90x + 200)

After simplifying further, you end up with your final answer:

(x⁴ - 2x³ - 33x² + 50x + 200)