Chapter 5.7, Problem 15WE

To determine

To factor: the given polynomial $42-23k+k^2$ and check the factors by multiplying.

Answer to Problem 15WE

The factors of $k^2-23k+42$ are $\left(k-2\right)\left(k-21\right)$ .

Explanation of Solution

Given information: $42-23k+k^2$ .

Calculations: We have given a quadratic $42-23k+k^2$ .

Arrange the given quadratic in standard form $a{x}^2+bx+c$ , we get

  $k^2-23k+42$

Now, we need to compare given quadratic with standard quadratic $a{x}^2+bx+c$ and get the values of a, b and c . Then, apply product sum rule to factor the quadratic.

On comparing given quadratic $k^2-23k+42$ with standard quadratic $a{x}^2+bx+c$ , we get

  $a=1,b=-23andc=42.$

Product of value of a and c = $1\times 42=42$ .

Value of $b=-23.$

Now, we need to get two factors of 42 those add upto $-23$ .

We could see $42=-21\cdot \left(-2\right)\text{ and }-23=-21+\left(-2\right)$

So, split middle term $16x$ into $-2k-21k$ , we get

  $k^2-2k-21k+42$

Now, making it into two groups and factor out Greatest Common Factor (GCF) of each group

  $\left(k^2-2k\right)+\left(-21k+42\right)$

Factor out Greatest Common Factor (GCF) $k$ from $\left(k^2-2k\right)$ and $-21$ from $\left(-21k+42\right)$

  $k\left(k-2\right)-21\left(k-2\right)$

Factor out the common parenthesis $(k-2)$

  $\left(k-2\right)\left(k-21\right)$

Now, let us check it’s factor by multiplying.

In order to multiply two factors of a quadratic, we could apply First Outer Inner Last (FOIL) method.

Therefore,

  $\begin{array}{l}\left(k-2\right)\left(k-21\right)=k\cdot k+\text{ k}\cdot -21-2\cdot k-2\cdot -21\\ \text{ First Out Inner Last}\end{array}$

Simplify now, we get

  $k^2-21k-2k+42$

Combine like terms

  $k^2-23k+42$

We got same polynomial as it was given.

Therefore, the factors of $k^2-23k+42$ are $\left(k-2\right)\left(k-21\right)$ .