High School

Complete the synthetic division problem below:

[tex]-1 \longdiv {275}[/tex]

What is the quotient in polynomial form?

A. [tex]2x + 5[/tex]
B. [tex]x - 5[/tex]
C. [tex]2x - 5[/tex]
D. [tex]x + 5[/tex]

Answer :

To solve this synthetic division problem, you are dividing the polynomial [tex]\(2x^2 + 7x + 5\)[/tex] by the binomial [tex]\(x + 1\)[/tex].

Here’s a step-by-step explanation of how the synthetic division works for this:

1. Identify the coefficients: The polynomial [tex]\(2x^2 + 7x + 5\)[/tex] has the coefficients [2, 7, 5].

2. Set up synthetic division: The divisor [tex]\(x + 1\)[/tex] means you will use -1 for synthetic division. This is because we use the root of the divisor, which is -1 in this case (since [tex]\(x + 1 = x - (-1)\)[/tex]).

3. Start the synthetic division process:
- Write -1 on the left and the coefficients [2, 7, 5] on the right.

- Bring down the first coefficient, which is 2.

4. Perform the process:
- Multiply -1 by the first coefficient 2 and write the result under the next coefficient:
[tex]\[
-1 \times 2 = -2
\][/tex]
- Add this result to 7 (the next coefficient):
[tex]\[
7 + (-2) = 5
\][/tex]
- Write 5 below the line.

- Multiply -1 by 5 (the new number just written) and write under the next coefficient:
[tex]\[
-1 \times 5 = -5
\][/tex]
- Add this to 5 (the last coefficient):
[tex]\[
5 + (-5) = 0
\][/tex]
- Write 0 below the line, which confirms there's a remainder of 0.

5. Form the quotient: The resulting numbers above the line, [2, 5], give you the coefficients of the quotient polynomial. The quotient of the division [tex]\(2x^2 + 7x + 5\)[/tex] by [tex]\(x + 1\)[/tex] is [tex]\(2x + 5\)[/tex].

Therefore, the polynomial quotient is [tex]\(2x + 5\)[/tex].

The correct answer is:

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