Two roots of the polynomial function f(x) = x3 − 7x − 6 are −2 and 3. Use the fundamental theorem of algebra and the complex con

Question

kolezko

3 months ago

13

Two roots of the polynomial function f(x) = x3 − 7x − 6 are −2 and 3. Use the fundamental theorem of algebra and the complex con

jugate theorem to determine the number and nature of the remaining root(s). Explain your thinking.

Mathematics

2 answers:

lawyer [12.5K] · Answer 1 · 2026-02-19T16:51:27+03:00

Answer:

-1

Step-by-step explanation:

The roots of the polynomial are -2 and 3. According to the zero product property:

$x+2=0\\\\x-3=0 \\\\(x+2)(x-3)(x+a)=x^3-7x-6$

Given that the last term in the cubic equation is -6, it indicates that a must equal 1 as the product of -2 and 3 already results in -6. This can be verified by expanding the three factors:

$(x+2)(x-3)(x+1)=x^3+x^2-3x^2-3x+2x^2+2x-6x-6=x^3-7x-6$

Thus, following the zero product property, the final value is -1.

Hope this clarifies!

lawyer [12.5K] · Answer 2 · 2026-02-19T16:54:59+03:00

Answer:

The polynomial has a degree of 3.

According to the fundamental theorem of algebra, there are three roots for the function.

With two roots provided, it follows that one root remains.

As per the complex conjugate theorem, if there are any imaginary roots, they will appear in pairs.

This means the remaining root must be real.

Step-by-step explanation: