Answer :
To perform synthetic division and find the quotient in polynomial form, follow these steps:
1. Understand the Problem: You have a polynomial represented by its coefficients: 2 (for [tex]\(2x^2\)[/tex]), 7 (for [tex]\(7x\)[/tex]), and 5 (for the constant term). You're dividing by [tex]\(x + 1\)[/tex], which implies the divisor is [tex]\(-1\)[/tex].
2. Set Up Synthetic Division: Write down the coefficients: 2, 7, and 5. The divisor for synthetic division is the opposite of what is in the binomial divisor [tex]\(x + 1\)[/tex], so use [tex]\(-1\)[/tex].
3. Process the Synthetic Division:
- Start with the leading coefficient: 2. Write it down as it is.
- Multiply this result (2) by the divisor ([tex]\(-1\)[/tex]) and write below the next coefficient (7).
- [tex]\(2 \times (-1) = -2\)[/tex]
- Add the result [tex]\(-2\)[/tex] to the next coefficient:
- [tex]\(7 + (-2) = 5\)[/tex]
- Write this result below the 7.
- Repeat the process:
- Multiply the new result (5) by the divisor ([tex]\(-1\)[/tex]) and add to the last coefficient (5).
- [tex]\(5 \times (-1) = -5\)[/tex]
- Add this result to the final coefficient:
- [tex]\(5 + (-5) = 0\)[/tex]
4. Interpret the Results:
- The numbers you have now are: 2, 5, and the remainder is 0.
- The result corresponds to the coefficients of the quotient polynomial. Since you started with [tex]\(x^2\)[/tex], the quotient is one degree less, namely a linear polynomial:
- The quotient is [tex]\(2x + 5\)[/tex].
- Since the remainder is 0, there is no leftover from the division.
5. Choose the Correct Answer:
- From the options provided, the correct polynomial form of the quotient is [tex]\(2x + 5\)[/tex].
So, the correct answer is B. [tex]\(2x + 5\)[/tex].
1. Understand the Problem: You have a polynomial represented by its coefficients: 2 (for [tex]\(2x^2\)[/tex]), 7 (for [tex]\(7x\)[/tex]), and 5 (for the constant term). You're dividing by [tex]\(x + 1\)[/tex], which implies the divisor is [tex]\(-1\)[/tex].
2. Set Up Synthetic Division: Write down the coefficients: 2, 7, and 5. The divisor for synthetic division is the opposite of what is in the binomial divisor [tex]\(x + 1\)[/tex], so use [tex]\(-1\)[/tex].
3. Process the Synthetic Division:
- Start with the leading coefficient: 2. Write it down as it is.
- Multiply this result (2) by the divisor ([tex]\(-1\)[/tex]) and write below the next coefficient (7).
- [tex]\(2 \times (-1) = -2\)[/tex]
- Add the result [tex]\(-2\)[/tex] to the next coefficient:
- [tex]\(7 + (-2) = 5\)[/tex]
- Write this result below the 7.
- Repeat the process:
- Multiply the new result (5) by the divisor ([tex]\(-1\)[/tex]) and add to the last coefficient (5).
- [tex]\(5 \times (-1) = -5\)[/tex]
- Add this result to the final coefficient:
- [tex]\(5 + (-5) = 0\)[/tex]
4. Interpret the Results:
- The numbers you have now are: 2, 5, and the remainder is 0.
- The result corresponds to the coefficients of the quotient polynomial. Since you started with [tex]\(x^2\)[/tex], the quotient is one degree less, namely a linear polynomial:
- The quotient is [tex]\(2x + 5\)[/tex].
- Since the remainder is 0, there is no leftover from the division.
5. Choose the Correct Answer:
- From the options provided, the correct polynomial form of the quotient is [tex]\(2x + 5\)[/tex].
So, the correct answer is B. [tex]\(2x + 5\)[/tex].
