Answer :
Sure! Let's solve the synthetic division problem step-by-step.
We are given a polynomial with coefficients [tex]\(2, 27, 5\)[/tex] and we need to use synthetic division to divide it by [tex]\(x + 1\)[/tex], which means we will use [tex]\(-1\)[/tex] for the synthetic division process.
Here are the steps to complete the synthetic division:
1. Write down the coefficients of the polynomial:
- The polynomial is written in terms of [tex]\(x\)[/tex]. The coefficients are [tex]\(2\)[/tex], [tex]\(27\)[/tex], and [tex]\(5\)[/tex].
2. Set up the synthetic division:
- We use the value [tex]\(-1\)[/tex] (the opposite sign of the divisor [tex]\(x + 1\)[/tex]).
3. Carry down the leading coefficient:
- Start by writing the leading coefficient in the first position below the line.
4. Perform the synthetic division steps:
[tex]\[
\begin{array}{r|rrr}
-1 & 2 & 27 & 5 \\
& & 2 & 25 & \\
\hline
& 2 & 25 & -20 \\
\end{array}
\][/tex]
Here's how we calculate each step:
- Bring down the first coefficient: [tex]\(2\)[/tex].
- Multiply [tex]\(2\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-2\)[/tex], then add it to the second coefficient: [tex]\(27 + (-2) = 25\)[/tex].
- Multiply [tex]\(25\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-25\)[/tex], then add it to the third coefficient: [tex]\(5 + (-25) = -20\)[/tex].
5. Interpret the results:
- The quotient comes from the numbers [tex]\(2\)[/tex] and [tex]\(25\)[/tex] (ignoring the remainder, [tex]\(-20\)[/tex]).
- This represents the polynomial [tex]\(2x + 25\)[/tex].
Given the options provided:
A. [tex]\(2x + 5\)[/tex]
B. [tex]\(x + 5\)[/tex]
C. [tex]\(x - 5\)[/tex]
D. [tex]\(2x - 5\)[/tex]
None of these match the quotient [tex]\(2x + 25\)[/tex]. Therefore, the correct choice is not listed in the options.
So, based on the synthetic division process, the quotient in polynomial form is [tex]\(2x + 25\)[/tex], which is not given in the provided options.
We are given a polynomial with coefficients [tex]\(2, 27, 5\)[/tex] and we need to use synthetic division to divide it by [tex]\(x + 1\)[/tex], which means we will use [tex]\(-1\)[/tex] for the synthetic division process.
Here are the steps to complete the synthetic division:
1. Write down the coefficients of the polynomial:
- The polynomial is written in terms of [tex]\(x\)[/tex]. The coefficients are [tex]\(2\)[/tex], [tex]\(27\)[/tex], and [tex]\(5\)[/tex].
2. Set up the synthetic division:
- We use the value [tex]\(-1\)[/tex] (the opposite sign of the divisor [tex]\(x + 1\)[/tex]).
3. Carry down the leading coefficient:
- Start by writing the leading coefficient in the first position below the line.
4. Perform the synthetic division steps:
[tex]\[
\begin{array}{r|rrr}
-1 & 2 & 27 & 5 \\
& & 2 & 25 & \\
\hline
& 2 & 25 & -20 \\
\end{array}
\][/tex]
Here's how we calculate each step:
- Bring down the first coefficient: [tex]\(2\)[/tex].
- Multiply [tex]\(2\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-2\)[/tex], then add it to the second coefficient: [tex]\(27 + (-2) = 25\)[/tex].
- Multiply [tex]\(25\)[/tex] by [tex]\(-1\)[/tex] to get [tex]\(-25\)[/tex], then add it to the third coefficient: [tex]\(5 + (-25) = -20\)[/tex].
5. Interpret the results:
- The quotient comes from the numbers [tex]\(2\)[/tex] and [tex]\(25\)[/tex] (ignoring the remainder, [tex]\(-20\)[/tex]).
- This represents the polynomial [tex]\(2x + 25\)[/tex].
Given the options provided:
A. [tex]\(2x + 5\)[/tex]
B. [tex]\(x + 5\)[/tex]
C. [tex]\(x - 5\)[/tex]
D. [tex]\(2x - 5\)[/tex]
None of these match the quotient [tex]\(2x + 25\)[/tex]. Therefore, the correct choice is not listed in the options.
So, based on the synthetic division process, the quotient in polynomial form is [tex]\(2x + 25\)[/tex], which is not given in the provided options.
