A student claims that -9i is the only imaginary root of a quadratic polynomial equation that has real coefficients. a. What is the mistake in the student's claim? HINT: If there is another solution, what is it. If there is another solution, what theorem is used to support your claim. b. Write one possible factored polynomial that has the correct roots from part a by filling in the missing parts to the polynomial below. C. Write the factored polynomial from part b in standard form.

Question

A student claims that -9i is the only imaginary root of a quadratic polynomial equation that has real coefficients. a. What is the mistake in the student's claim? HINT: If there is another solution, what is it. If there is another solution, what theorem is used to support your claim. b. Write one possible factored polynomial that has the correct roots from part a by filling in the missing parts to the polynomial below. C. Write the factored polynomial from part b in standard form.

Answer 1

a. The mistake in the student's claim is that every imaginary root of a quadratic polynomial with real coefficients must come in conjugate pairs. Therefore, if -9i is a root, then its conjugate, 9i, must also be a root. This is because if the coefficients of the polynomial are real, then the complex conjugate of any complex number must also be a root of the polynomial.

b. One possible factored polynomial that has the correct roots is $(x + 9i)(x - 9i)$.

c. To write the factored polynomial from part b in standard form, we can multiply it out:
$(x + 9i)(x - 9i) = x^2 - 9ix + 9ix - 81i^2 = x^2 - 81i^2$.

Since $i^2$ is equal to -1, the standard form of the factored polynomial is $x^2 + 81$.