Chapter 8, Problem 49SGR

To determine

To factorize the given polynomial

Answer to Problem 49SGR

  $(x-5)(x-3)$

Explanation of Solution

Given:

Polynomial: $x^2-8x+15$

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,

  $\Rightarrow x^2-8x+15$

Substituting in the above formula where $a=1,b=-8,c=15$

  $\begin{array}{l}\Rightarrow x=\frac{-(-8)\pm \sqrt{{(-8)}^2-4(1)(15)}}{2(1)}\\ \Rightarrow x=\frac{8\pm \sqrt{64-60}}{2}\\ \Rightarrow x=\frac{8\pm \sqrt{4}}{2}\\ \Rightarrow x=\frac{8\pm 2}{2}\end{array}$

The roots are,

  $\begin{array}{l}\Rightarrow x_1=\frac{8+2}{2}=\frac{10}{2}=5\\ \Rightarrow x_2=\frac{8-2}{2}=\frac{6}{2}=3\end{array}$

The roots of the quadratic polynomial $x^2-8x+15=(x-5)(x-3).$

Conclusion:

Therefore, the factors of the polynomial are $(x-5)(x-3).$