Mastery Check Solutions

Show What You Know

Use synthetic division to find the missing value.

Note

Problems A–C
The highlighted terms should be placed into the synthetic division problem first. 

Find the value of n so that the remainder is 8 when $(x^{3} + 5 x^{2} + 12 x + n) \div (x + 4)$

$\begin{array}{rcl} x + 4 & = & 0 \\ x & = & - 4 \end{array}$

$\begin{array}{rcl} n + (- 32) & = & 8 \\ n & = & 40 \end{array}$

Note

Q: Where is a given remainder placed in a division problem?
A: The bottom right corner 

Note

Problems B–C
Because the constant and the remainder are both known in the right-most column, the missing value can be determined without the variable. 

Find the value of n so that the remainder is 3 when $(4 x^{2} + n x - 7) {(x - 1)}^{- 1}$

$\begin{array}{rcl} x - 1 & = & 0 \\ x & = & 1 \end{array}$

$\begin{array}{rcl} - 7 + n + 4 & = & 3 \\ - 3 + n & = & 3 \\ n & = & 6 \end{array}$

Find the value of n so that the remainder is –17 when $(x^{2} - 5 x - 11) \div (x - n)$

$\begin{array}{rcl} x - n & = & 0 \\ x & = & n \end{array}$

Solve for n using: $- 11 + n (n - 5) = - 17$ .
$\begin{array}{rcl} n^{2} - 5 n & = & - 6 \\ n^{2} - 5 n + 6 & = & 0 \\ (n - 2) (n - 3) & = & 0 \\ n & = & 2, 3 \end{array}$

Note

Q: Why does the expression need to be factored in order to be solved?
A: Because the expression is a quadratic

Say What You Know

In your own words, talk about what you have learned using the objectives for this part of the lesson and your work on this page.

Note

Restate the objectives of the lesson in your own words. If you are unable to restate the lesson objectives, go back and reread the objectives and then explain them.

Use synthetic division with polynomial expressions.
Evaluate polynomials using synthetic substitution.
Use the Remainder Theorem to prove polynomial division is correct