Answer :
Sure! Let's work through this synthetic division problem step-by-step.
We are given the polynomial [tex]\( x^2 + 5x - 14 \)[/tex] and are using synthetic division to divide it by [tex]\( x - 2 \)[/tex].
The coefficients of the polynomial are: [tex]\( 1 \)[/tex] (for [tex]\( x^2 \)[/tex]), [tex]\( 5 \)[/tex] (for [tex]\( x \)[/tex]), and [tex]\( -14 \)[/tex] (the constant term).
The divisor here is [tex]\( x - 2 \)[/tex], which means we'll be using the value [tex]\( 2 \)[/tex] in our synthetic division.
1. Setup: Write down the coefficients of the polynomial:
```
1 5 -14
```
Next, we use the value [tex]\( 2 \)[/tex] (since we're dividing by [tex]\( x - 2 \)[/tex]).
2. Process: Start with the leading coefficient:
- Bring down the [tex]\( 1 \)[/tex] as is.
3. Multiply and Add:
- Multiply the [tex]\( 1 \)[/tex] by [tex]\( 2 \)[/tex] (our synthetic divisor), which gives [tex]\( 2 \)[/tex].
- Add this [tex]\( 2 \)[/tex] to the next coefficient (which is [tex]\( 5 \)[/tex]):
[tex]\[
5 + 2 = 7
\][/tex]
4. Repeat:
- Multiply the [tex]\( 7 \)[/tex] by [tex]\( 2 \)[/tex] (our synthetic divisor), which gives [tex]\( 14 \)[/tex].
- Add this [tex]\( 14 \)[/tex] to the last coefficient (which is [tex]\(-14\)[/tex]):
[tex]\[
-14 + 14 = 0
\][/tex]
5. Result: The numbers left at the bottom are the coefficients of the quotient. In this case, we're left with:
```
1 7
```
The last number, [tex]\( 0 \)[/tex], is the remainder.
Thus, the quotient polynomial is [tex]\( x + 7 \)[/tex].
So, the resulting polynomial is:
[tex]\[ x + 7 \][/tex]
Therefore, the correct answer is A. [tex]\( x + 7 \)[/tex].
We are given the polynomial [tex]\( x^2 + 5x - 14 \)[/tex] and are using synthetic division to divide it by [tex]\( x - 2 \)[/tex].
The coefficients of the polynomial are: [tex]\( 1 \)[/tex] (for [tex]\( x^2 \)[/tex]), [tex]\( 5 \)[/tex] (for [tex]\( x \)[/tex]), and [tex]\( -14 \)[/tex] (the constant term).
The divisor here is [tex]\( x - 2 \)[/tex], which means we'll be using the value [tex]\( 2 \)[/tex] in our synthetic division.
1. Setup: Write down the coefficients of the polynomial:
```
1 5 -14
```
Next, we use the value [tex]\( 2 \)[/tex] (since we're dividing by [tex]\( x - 2 \)[/tex]).
2. Process: Start with the leading coefficient:
- Bring down the [tex]\( 1 \)[/tex] as is.
3. Multiply and Add:
- Multiply the [tex]\( 1 \)[/tex] by [tex]\( 2 \)[/tex] (our synthetic divisor), which gives [tex]\( 2 \)[/tex].
- Add this [tex]\( 2 \)[/tex] to the next coefficient (which is [tex]\( 5 \)[/tex]):
[tex]\[
5 + 2 = 7
\][/tex]
4. Repeat:
- Multiply the [tex]\( 7 \)[/tex] by [tex]\( 2 \)[/tex] (our synthetic divisor), which gives [tex]\( 14 \)[/tex].
- Add this [tex]\( 14 \)[/tex] to the last coefficient (which is [tex]\(-14\)[/tex]):
[tex]\[
-14 + 14 = 0
\][/tex]
5. Result: The numbers left at the bottom are the coefficients of the quotient. In this case, we're left with:
```
1 7
```
The last number, [tex]\( 0 \)[/tex], is the remainder.
Thus, the quotient polynomial is [tex]\( x + 7 \)[/tex].
So, the resulting polynomial is:
[tex]\[ x + 7 \][/tex]
Therefore, the correct answer is A. [tex]\( x + 7 \)[/tex].
