Question

Two roots of the polynomial function f(x) = x3 − 7x − 6 are −2 and 3. Use the fundamental theorem of algebra and the complex conjugate theorem to determine the number and nature of the remaining root(s). Explain your thinking.

Answer 1

Answer:

-1

Step-by-step explanation:

If the roots are -2 and 3, then by the zero product rule:

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

Since the last term of the cubic equation is -6, a must be 1 since the product of 2 and -3 is already -6. This can be confirmed by multiplying out the three terms:

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

Therefore, by the zero product rule, the last rule is -1.

Hope this helps!

Answer 2

Answer:

The degree of the polynomial is 3.

By the fundamental theorem of algebra, the function has three roots.

Two roots are given, so there must be one root remaining.

By the complex conjugate theorem, imaginary roots come in pairs.

The final root must be real.

Step-by-step explanation: