High School

Complete the synthetic division problem below.

[tex]\[

-1 \longdiv \begin{array}{c|ccc}

& 2 & 7 & 5 \\

\end{array}

\][/tex]

What is the quotient in polynomial form?

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

B. [tex]\( 2x + 5 \)[/tex]

C. [tex]\( 2x - 5 \)[/tex]

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

Answer :

Sure! Let's go through the process of synthetic division step-by-step to find the quotient when dividing the polynomial [tex]\( 2x^2 + 7x + 5 \)[/tex] by [tex]\( x + 1 \)[/tex].

1. Set up the synthetic division:
Since we are dividing by [tex]\( x + 1 \)[/tex], we will use the root [tex]\( -1 \)[/tex] for synthetic division.

2. Write down the coefficients:
We take the coefficients of the polynomial [tex]\( 2x^2 + 7x + 5 \)[/tex], which are 2, 7, and 5.

3. Begin synthetic division:
- Start with the first coefficient, which is 2. Bring it down as is:
[tex]\( 2 \)[/tex].

4. Multiply and add:
- Multiply the root (-1) by 2, the number we just brought down, giving us:
[tex]\( -1 \times 2 = -2 \)[/tex].
- Add this result to the next coefficient (7):
[tex]\( 7 + (-2) = 5 \)[/tex].
- Write down this new number (5) below the line.

5. Continue multiplying and adding:
- Multiply the root (-1) by the number you just obtained (5):
[tex]\( -1 \times 5 = -5 \)[/tex].
- Add this result to the last coefficient (5):
[tex]\( 5 + (-5) = 0 \)[/tex].
- This value (0) is the remainder.

6. Construct the quotient polynomial:
The numbers you have written below the line, excluding the remainder, form the coefficients of the quotient. So, the quotient is [tex]\( 2x + 5 \)[/tex].

Therefore, the quotient from the synthetic division is [tex]\( 2x + 5 \)[/tex]. The correct option is:

B. [tex]\( 2x + 5 \)[/tex].