Chapter 8.6, Problem 1P

a.

To determine

To calculate: The factored form the given expression

Answer to Problem 1P

  $\left(3x+5\right)\left(2x+1\right)$

Explanation of Solution

Given: Expression is given as $6x^2+13x+5$

Calculation:

Since the quadratic expression is in the form $a{x}^2+bx+c$ , therefore we use grouping method which is most applicable method to be used here and is solved as following-

  $6x^2+13x+5$

Find the factor of $6\times 5$ which result into as $13$ after adding

and by inspection 10 and $3$ is the answer

therefore,split the middle term

  $\Rightarrow 6x^2+10x+3x+5$

Factor out greatest common factor and then factor out $\left(3x+5\right)$

  $\begin{array}{l}\Rightarrow 2x\left(3x+5\right)+1\left(3x+5\right)\\ \Rightarrow \left(3x+5\right)\left(2x+1\right)\end{array}$

b.

To determine

To show: explaining the reason about the factors of $ac$ of the expression.

Explanation of Solution

Given information: the expression is given as $a{x}^2+bx+c$ where $ac$ is positive and $b$ is negative.

Verification:

An expression is in factored form only if its expression specify its product.Factoring is a useful tool in solving higher degree equations as it can  change an expression from a sum or difference of terms to a product of factors.

The given polynomial is of the form $a{x}^2+bx+c$ which can be factored only if there are two factors of $ac$ and that will be the sum of $b$ .Here the best applicable method to be used is grouping method and is can be better explained by the following example-

  $3x^2-7x+4$

Find the factor of $3\times 4$ which result into as $12$ after adding

and by inspection -3 and -4 is the answer

therefore, split the middle term

  $\Rightarrow 3x^2-4x-3x+12$

Factor out greatest common factor and then factor out $\left(x-4\right)$

  $\begin{array}{l}\Rightarrow 3x\left(x-4\right)-3\left(x-4\right)\\ \Rightarrow \left(x-4\right)\left(3x-3\right)\end{array}$

Take common factor 3 out

  $\Rightarrow 3\left(x-4\right)\left(x-1\right)$

Hence from the above example we can define the factor of $ac$ is $12$ and that of b is $-7$