Chapter 8, Problem 53SGR

To determine

To factorize the given polynomial

Answer to Problem 53SGR

  $(x-5)(x+10)$

Explanation of Solution

Given:

Polynomial: $x^2+5x-50$

Formula used:

Roots of the polynomial of the form $a{x}^2+bx+c=a(x-x_1)(x-x_2)$ are

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Consider the given polynomial,

  $\Rightarrow x^2+5x-50$

Substituting in the above formula where $a=1,b=5,c=-50$

  $\begin{array}{l}\Rightarrow x=\frac{-5\pm \sqrt{5^2-4(1)(-50)}}{2(1)}\\ \Rightarrow x=\frac{-5\pm \sqrt{25+200}}{2}\\ \Rightarrow x=\frac{-5\pm \sqrt{225}}{2}\\ \Rightarrow x=\frac{-5\pm 15}{2}\end{array}$

The roots are,

  $\begin{array}{l}\Rightarrow x_1=\frac{-5+15}{2}=\frac{10}{2}=5\\ \Rightarrow x_2=\frac{-5-15}{2}=\frac{-20}{2}=-10\end{array}$

The roots of the quadratic polynomial $x^2+5x-50=(x-5)(x+10).$

Conclusion:

Therefore, the factors of the polynomial are $(x-5)(x+10).$