Question

Application/Communication: Create a two-term and a three-term quadratic polynomial which cannot be factored and explain why factoring is impossible.

Answer 1

ANSWER:

a) two-term quadratic polynomial

$f(x)=x^2+9$

b) three-term quadratic polynomial

$f(x)=x^2+2x+15$

STEP-BY-STEP EXPLANATION:

a) two-term quadratic polynomial

$f(x)=x^2+9$

Here factoring is impossible because, then f(x) = 0:

$\begin{gathered} x^2+9=0 \\ \\ x^2=-9 \\ \\ x=\pm\sqrt{-9}\rightarrow\text{ imaginary roots} \end{gathered}$

Factoring is only possible when the polynomials have real roots.

Since value of x is imaginary, factoring is impossible

b) three-term quadratic polynomial

$f(x)=x^2+2x+15$

Here also when f(x) = 0

$\begin{gathered} x=\frac{-2\pm\sqrt{2^2-4(1)(15)}}{2(1)} \\ \\ x=\frac{-2\pm\sqrt{2^2-60}}{2}\: \\ \\ x=\frac{-2\pm\sqrt{-56}}{2}\:\rightarrow\text{ complex roots} \end{gathered}$

Therefore, factoring in this case is also not possible.