Question

Given that 2i is one of the zeros of p(x)=x⁴−5x³+8x²−20x+16, find the other zeros. Use synthetic division and then write the linear factorization of p(x).

A) (x−2i)(x³−2x²+3x−6)
B) (x+2i)(x³−2x²+3x−6)
C) (x−2i)(x³+2x²+3x−6)
D) (x+2i)(x³+2x²+3x−6)

Answer 1

The zeros of the polynomial p(x)=x⁴−5x³+8x²−20x+16 include 2i and its conjugate −2i. By using synthetic division and the quadratic formula, the remaining zeros can be found, allowing for the construction of the linear factorization of p(x).

Step-by-step explanation:

Since 2i is a zero of the polynomial p(x)=x⁴−5x³+8x²−20x+16, its conjugate −2i must also be a zero because the coefficients of the polynomial are all real numbers. Using synthetic division, we can find the quotient polynomial when p(x) is divided by (x−2i)(x+2i), which simplifies to (x2+4).

First, synthetic division of p(x) by x2+4 yields the quadratic polynomial q(x)=x²−5x+4. Next, we factorize or use the quadratic formula to find the zeros of q(x). The solutions to the quadratic equation ax2+bx+c=0 can be found using the formula x = −b±√(b2−4ac)/(2a), which gives us the remaining two zeros of the original polynomial.

After finding all the zeros and considering the multiplicity of each zero, we can write the linear factorization of p(x). The answer to the student's question will be one of the given multiple-choice options, reflecting the product of linear factors corresponding to each zero.