College

Complete the synthetic division problem below.

[tex]\[ 2 \longdiv 1 \quad 5 \quad -1 \quad 4 \][/tex]

What is the quotient in polynomial form?

A. [tex]\( x - 7 \)[/tex]

B. [tex]\( x + 7 \)[/tex]

C. [tex]\( x + 5 \)[/tex]

D. [tex]\( x - 5 \)[/tex]

Answer :

We are given the dividend polynomial

[tex]$$
x^2 + 5x - 14
$$[/tex]

and we want to divide it by

[tex]$$
x - 2.
$$[/tex]

Since the divisor is linear, synthetic division is a good method.

Step 1: Write down the coefficients of the dividend polynomial:

[tex]$$
[1,\ 5,\ -14].
$$[/tex]

Step 2: Write the synthetic divisor, which is the zero of [tex]$x-2$[/tex], so it is [tex]$2$[/tex].

Step 3: Begin the synthetic division process:

1. Bring down the first coefficient: [tex]$1$[/tex].

2. Multiply [tex]$1$[/tex] by the divisor [tex]$2$[/tex] to get [tex]$2$[/tex]. Write this beneath the next coefficient.

3. Add [tex]$2$[/tex] to the second coefficient [tex]$5$[/tex] to get [tex]$7$[/tex].

4. Multiply [tex]$7$[/tex] by the divisor [tex]$2$[/tex] to get [tex]$14$[/tex]. Write this beneath the third coefficient.

5. Add [tex]$14$[/tex] to the third coefficient [tex]$-14$[/tex] to get [tex]$0$[/tex], which is our remainder.

So, the synthetic division process yields the numbers:

[tex]$$
\text{Quotient coefficients: } [1,\ 7] \quad \text{and Remainder: } 0.
$$[/tex]

Step 4: Interpret the quotient coefficients. The first coefficient corresponds to the term with [tex]$x$[/tex] and the second as the constant:

[tex]$$
x + 7.
$$[/tex]

Thus, the quotient in polynomial form is

[tex]$$
\boxed{x + 7}.
$$[/tex]

This result corresponds to option B: [tex]$x+7$[/tex].