Answer :
We are given the dividend polynomial
[tex]$$
x^2 + 5x - 14
$$[/tex]
and we want to divide it by
[tex]$$
x - 2.
$$[/tex]
Since the divisor is linear, synthetic division is a good method.
Step 1: Write down the coefficients of the dividend polynomial:
[tex]$$
[1,\ 5,\ -14].
$$[/tex]
Step 2: Write the synthetic divisor, which is the zero of [tex]$x-2$[/tex], so it is [tex]$2$[/tex].
Step 3: Begin the synthetic division process:
1. Bring down the first coefficient: [tex]$1$[/tex].
2. Multiply [tex]$1$[/tex] by the divisor [tex]$2$[/tex] to get [tex]$2$[/tex]. Write this beneath the next coefficient.
3. Add [tex]$2$[/tex] to the second coefficient [tex]$5$[/tex] to get [tex]$7$[/tex].
4. Multiply [tex]$7$[/tex] by the divisor [tex]$2$[/tex] to get [tex]$14$[/tex]. Write this beneath the third coefficient.
5. Add [tex]$14$[/tex] to the third coefficient [tex]$-14$[/tex] to get [tex]$0$[/tex], which is our remainder.
So, the synthetic division process yields the numbers:
[tex]$$
\text{Quotient coefficients: } [1,\ 7] \quad \text{and Remainder: } 0.
$$[/tex]
Step 4: Interpret the quotient coefficients. The first coefficient corresponds to the term with [tex]$x$[/tex] and the second as the constant:
[tex]$$
x + 7.
$$[/tex]
Thus, the quotient in polynomial form is
[tex]$$
\boxed{x + 7}.
$$[/tex]
This result corresponds to option B: [tex]$x+7$[/tex].
[tex]$$
x^2 + 5x - 14
$$[/tex]
and we want to divide it by
[tex]$$
x - 2.
$$[/tex]
Since the divisor is linear, synthetic division is a good method.
Step 1: Write down the coefficients of the dividend polynomial:
[tex]$$
[1,\ 5,\ -14].
$$[/tex]
Step 2: Write the synthetic divisor, which is the zero of [tex]$x-2$[/tex], so it is [tex]$2$[/tex].
Step 3: Begin the synthetic division process:
1. Bring down the first coefficient: [tex]$1$[/tex].
2. Multiply [tex]$1$[/tex] by the divisor [tex]$2$[/tex] to get [tex]$2$[/tex]. Write this beneath the next coefficient.
3. Add [tex]$2$[/tex] to the second coefficient [tex]$5$[/tex] to get [tex]$7$[/tex].
4. Multiply [tex]$7$[/tex] by the divisor [tex]$2$[/tex] to get [tex]$14$[/tex]. Write this beneath the third coefficient.
5. Add [tex]$14$[/tex] to the third coefficient [tex]$-14$[/tex] to get [tex]$0$[/tex], which is our remainder.
So, the synthetic division process yields the numbers:
[tex]$$
\text{Quotient coefficients: } [1,\ 7] \quad \text{and Remainder: } 0.
$$[/tex]
Step 4: Interpret the quotient coefficients. The first coefficient corresponds to the term with [tex]$x$[/tex] and the second as the constant:
[tex]$$
x + 7.
$$[/tex]
Thus, the quotient in polynomial form is
[tex]$$
\boxed{x + 7}.
$$[/tex]
This result corresponds to option B: [tex]$x+7$[/tex].
