Answer :
Sure! Let's divide the polynomial [tex]\(5x^5 - 35x^4 + 135x^3 - 285x^2 - 72x + 48\)[/tex] by [tex]\(x - 4\)[/tex] using synthetic division:
1. Identify the coefficients: The polynomial is [tex]\(5x^5 - 35x^4 + 135x^3 - 285x^2 - 72x + 48\)[/tex], so the coefficients are [tex]\(5, -35, 135, -285, -72, 48\)[/tex].
2. Write the root of the divisor: Since we are dividing by [tex]\(x - 4\)[/tex], the root is [tex]\(4\)[/tex].
3. Set up synthetic division:
- Write the root [tex]\(4\)[/tex] on the left.
- Write the coefficients [tex]\(5, -35, 135, -285, -72, 48\)[/tex] on the right.
4. Perform synthetic division steps:
- Bring down the first coefficient, [tex]\(5\)[/tex], to the bottom row.
- Multiply this number by the root [tex]\(4\)[/tex] and write the result under the next coefficient.
- Add the next coefficient to this result and write the sum under the line.
Repeat these steps for each coefficient:
[tex]\[
\begin{array}{r|rrrrr}
4 & 5 & -35 & 135 & -285 & -72 & 48 \\
& & 20 & 20 & 620 & 13 & -48 \\
\hline
& 5 & -15 & 75 & 15 & -12 & 0 \\
\end{array}
\][/tex]
5. Read the result: The numbers on the bottom row, except the last one, are the coefficients of the quotient polynomial. The last number is the remainder.
- The quotient is [tex]\(5x^4 - 15x^3 + 75x^2 + 15x - 12\)[/tex].
- The remainder is [tex]\(0\)[/tex].
So, the division of the polynomials results in:
[tex]\[
5x^5 - 35x^4 + 135x^3 - 285x^2 - 72x + 48 = (x - 4)(5x^4 - 15x^3 + 75x^2 + 15x - 12)
\][/tex]
This means the remainder is [tex]\(0\)[/tex], indicating that [tex]\(x - 4\)[/tex] is a factor of the polynomial.
1. Identify the coefficients: The polynomial is [tex]\(5x^5 - 35x^4 + 135x^3 - 285x^2 - 72x + 48\)[/tex], so the coefficients are [tex]\(5, -35, 135, -285, -72, 48\)[/tex].
2. Write the root of the divisor: Since we are dividing by [tex]\(x - 4\)[/tex], the root is [tex]\(4\)[/tex].
3. Set up synthetic division:
- Write the root [tex]\(4\)[/tex] on the left.
- Write the coefficients [tex]\(5, -35, 135, -285, -72, 48\)[/tex] on the right.
4. Perform synthetic division steps:
- Bring down the first coefficient, [tex]\(5\)[/tex], to the bottom row.
- Multiply this number by the root [tex]\(4\)[/tex] and write the result under the next coefficient.
- Add the next coefficient to this result and write the sum under the line.
Repeat these steps for each coefficient:
[tex]\[
\begin{array}{r|rrrrr}
4 & 5 & -35 & 135 & -285 & -72 & 48 \\
& & 20 & 20 & 620 & 13 & -48 \\
\hline
& 5 & -15 & 75 & 15 & -12 & 0 \\
\end{array}
\][/tex]
5. Read the result: The numbers on the bottom row, except the last one, are the coefficients of the quotient polynomial. The last number is the remainder.
- The quotient is [tex]\(5x^4 - 15x^3 + 75x^2 + 15x - 12\)[/tex].
- The remainder is [tex]\(0\)[/tex].
So, the division of the polynomials results in:
[tex]\[
5x^5 - 35x^4 + 135x^3 - 285x^2 - 72x + 48 = (x - 4)(5x^4 - 15x^3 + 75x^2 + 15x - 12)
\][/tex]
This means the remainder is [tex]\(0\)[/tex], indicating that [tex]\(x - 4\)[/tex] is a factor of the polynomial.
