High School

Complete the synthetic division problem below:

[tex]\[ 2 \left\lvert\, \begin{array}{lll} 1 & 5 & -14 \end{array}\right. \][/tex]

What is the quotient in polynomial form?

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

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

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

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

Answer :

We want to divide the polynomial

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

by the binomial divisor corresponding to the number [tex]$2$[/tex]. The synthetic division setup is:

[tex]$$
2 \ \big| \ 1 \quad 5 \quad -14
$$[/tex]

Here’s the step-by-step process:

1. Bring Down the Leading Coefficient:
Write the first coefficient, [tex]$1$[/tex], below the line.

2. Multiply and Add (First Iteration):
Multiply the number you just wrote, [tex]$1$[/tex], by [tex]$2$[/tex] (the number on the left) to get [tex]$2$[/tex].
Then add this result to the next coefficient:
[tex]$$
5 + 2 = 7.
$$[/tex]

3. Multiply and Add (Second Iteration):
Multiply [tex]$7$[/tex] (the new number below the line) by [tex]$2$[/tex] to get [tex]$14$[/tex].
Add this to the third coefficient:
[tex]$$
-14 + 14 = 0.
$$[/tex]

4. Interpret the Result:
- The numbers written below the line (except the last one) represent the coefficients of the quotient polynomial.
- From our calculation, the quotient coefficients are [tex]$1$[/tex] and [tex]$7$[/tex], which gives the polynomial:
[tex]$$
x + 7.
$$[/tex]
- The final number, [tex]$0$[/tex], is the remainder.

Thus, the quotient in polynomial form is:

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

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