Chapter 8, Problem 4SGR

To determine

To state whether the statement is true or false

Answer to Problem 4SGR

False

Explanation of Solution

Given:

Statement: $(x+5)(x-2)$ is the factored form of $x^2-3x-10$ .

Concept 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 expression,

  $x^2-3x-10$

Substituting in the above formula for $a=1,b=-3,c=-10$

  $\begin{array}{l}x=\frac{-(-3)\pm \sqrt{{(-3)}^2-4(1)(-10)}}{2(1)}\\ x=\frac{3\pm \sqrt{9+40}}{2}\\ x=\frac{3\pm \sqrt{49}}{2}\\ x=\frac{3\pm 7}{2}\end{array}$

Thus the roots are,

  $\begin{array}{l}{x}_1=\frac{3+7}{2}=\frac{10}{2}=5\\ x_2=\frac{3-7}{2}=\frac{-4}{2}=-2\end{array}$

Therefore, the expression can be factored as,

  $\begin{array}{l}\Rightarrow x^2-3x-10=(x-(-2))(x-5)\\ \Rightarrow x^2-3x-10=(x+2)(x-5)\end{array}$

The signs of the factors have to be changed in order to make the statement true.

Conclusion:

Therefore, $(x+5)(x-2)$ is not the factored form of $x^2-3x-10$ .