Chapter 8.6, Problem 51PPE

a.

To determine

To find:x-intercepts

Answer to Problem 51PPE

  $\left(-3,0\right),\left(-2,0\right)$

Explanation of Solution

Given information: The graph of the function $y=x^2+5x+6$ is given

  

x-intercepts are the points on the graph where the graph cuts the x-axis.

The graph of $y=x^2+5x+6$ cuts the x-axis at points $\left(-3,0\right),\left(-2,0\right)$

So,

x-intercepts are $\left(-3,0\right),\left(-2,0\right)$

b.

To determine

To factor: $x^2+5x+6$

Answer to Problem 51PPE

  $\left(x+2\right)\left(x+3\right)$

Explanation of Solution

Calculation:

Consider $x^2+5x+6$

Here,

  $5x=2x+3x$

So,

  $x^2+5x+6=x^2+2x+3x+6$

Take $x$ common from terms $x^2,2x$ and take $3$ common from terms $3x,6$

Therefore,

  $\begin{array}{c}{x}^2+5x+6=x^2+2x+3x+6\\ =x\left(x+2\right)+3\left(x+2\right)\\ =\left(x+2\right)\left(x+3\right)\end{array}$

c.

To determine

To find:relationship between binomial factors found in part b. and the x-intercepts

Explanation of Solution

In part b., we get binomial factors as $\left(x+2\right),\left(x+3\right)$

Put each of the binomial factors equal to 0.

  $\begin{array}{l}x+2=0\Rightarrow x=-2\\ x+3=0\Rightarrow x=-3\end{array}$

So, as we put the binomial factors equal to 0, we get the x-intercepts