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

a) two-term quadratic polynomial

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

b) three-term quadratic polynomial

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

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√(-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=(-2\pm√(2^2-4(1)(15)))/(2(1)) \\ \\ x=(-2\pm√(2^2-60))/(2)\: \\ \\ x=(-2\pm√(-56))/(2)\:\rightarrow\text{ complex roots} \end{gathered}`

Therefore, factoring in this case is also not possible.