Find a polynomial function of degree 4 with integer coefficients, whose leading coefficient is equal to one, and that has the zeros and .

Question

Find a polynomial function of degree 4 with integer coefficients, whose leading coefficient is equal to one, and that has the zeros  and  .

Quizplus · Accepted Answer

If the zeros of the polynomial are $a$ and $b$, then the polynomial can be written as $(x-a)(x-b)(cx^2+dx+e)$. Since the leading coefficient is one, we can take $c=1$, and because the coefficients are integers, we know that the quadratic factor must also have integer coefficients. Using the two given zeros, the quadratic factor is $(x-11ebfdc1_ec2a_42b9_abe3_e5f0b4e5ed18_TBX8727_11)(x-11ebfdc1_ec2a_42ba_abe3_7f9ff32fe184_TBX8727_11)= (x-11ebfdc1_ec2a_42b9_abe3_e5f0b4e5ed18_TBX8727_11)(x-11ebfdc1_ec2a_42b9_abe3_7f9ff32fe184_TBX8727_11)$. The degree of this quadratic factor is 2, so it must be multiplied by an (x-d)(x-e) factor to obtain a degree 4 polynomial. Since we want integer coefficients, we can take d and e to be the conjugates of the given zeros, which are $11ebfdc1_ec2a_42b9_abe3_e5f0b4e5ed18_TBX8727_11$ and $11ebfdc1_ec2a_42ba_abe3_7f9ff32fe184_TBX8727_11$. Therefore, the answer is (C).

Find a Polynomial Function of Degree 4 with Integer Coefficients