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].
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].
