Answer :
Sure! I'll guide you through the long division process step-by-step for the polynomial division problem:
We have:
[tex]\[ 5x - 2 \longdiv {25x^3 + 7x^2 - 8x + 10} \][/tex]
Step 1: Divide the first term of the dividend by the first term of the divisor.
The first term of the dividend is [tex]\(25x^3\)[/tex], and the first term of the divisor is [tex]\(5x\)[/tex].
[tex]\[ \frac{25x^3}{5x} = 5x^2 \][/tex]
Step 2: Multiply the entire divisor by the result from step 1.
[tex]\[ 5x^2 \times (5x - 2) = 25x^3 - 10x^2 \][/tex]
Step 3: Subtract the result from the original polynomial.
Place the multiplication result below the corresponding terms in the original polynomial, and subtract:
[tex]\[
\begin{align*}
& \phantom{-}25x^3 + 7x^2 - 8x + 10 \\
- & (25x^3 - 10x^2) \\
\hline
& \phantom{-}17x^2 - 8x + 10
\end{align*}
\][/tex]
Step 4: Repeat the process with the new polynomial [tex]\(17x^2 - 8x + 10\)[/tex].
Divide the first term of the new polynomial by the first term of the divisor.
[tex]\[ \frac{17x^2}{5x} = \frac{17}{5}x \][/tex]
Step 5: Multiply the entire divisor by the result from the division.
[tex]\[ \frac{17}{5}x \times (5x - 2) = 17x^2 - \frac{34}{5}x \][/tex]
Step 6: Subtract this result from the current polynomial.
Place the multiplication result below and subtract:
[tex]\[
\begin{align*}
& \phantom{-}17x^2 - 8x + 10 \\
- & (17x^2 - \frac{34}{5}x) \\
\hline
& \phantom{-}-\frac{6}{5}x + 10
\end{align*}
\][/tex]
Continue the process until the degree of the remainder polynomial is less than the divisor or zero.
By following this approach, and filling in the blanks accordingly, you complete the division and solve the problem step-by-step. If you have any more questions, feel free to ask!
We have:
[tex]\[ 5x - 2 \longdiv {25x^3 + 7x^2 - 8x + 10} \][/tex]
Step 1: Divide the first term of the dividend by the first term of the divisor.
The first term of the dividend is [tex]\(25x^3\)[/tex], and the first term of the divisor is [tex]\(5x\)[/tex].
[tex]\[ \frac{25x^3}{5x} = 5x^2 \][/tex]
Step 2: Multiply the entire divisor by the result from step 1.
[tex]\[ 5x^2 \times (5x - 2) = 25x^3 - 10x^2 \][/tex]
Step 3: Subtract the result from the original polynomial.
Place the multiplication result below the corresponding terms in the original polynomial, and subtract:
[tex]\[
\begin{align*}
& \phantom{-}25x^3 + 7x^2 - 8x + 10 \\
- & (25x^3 - 10x^2) \\
\hline
& \phantom{-}17x^2 - 8x + 10
\end{align*}
\][/tex]
Step 4: Repeat the process with the new polynomial [tex]\(17x^2 - 8x + 10\)[/tex].
Divide the first term of the new polynomial by the first term of the divisor.
[tex]\[ \frac{17x^2}{5x} = \frac{17}{5}x \][/tex]
Step 5: Multiply the entire divisor by the result from the division.
[tex]\[ \frac{17}{5}x \times (5x - 2) = 17x^2 - \frac{34}{5}x \][/tex]
Step 6: Subtract this result from the current polynomial.
Place the multiplication result below and subtract:
[tex]\[
\begin{align*}
& \phantom{-}17x^2 - 8x + 10 \\
- & (17x^2 - \frac{34}{5}x) \\
\hline
& \phantom{-}-\frac{6}{5}x + 10
\end{align*}
\][/tex]
Continue the process until the degree of the remainder polynomial is less than the divisor or zero.
By following this approach, and filling in the blanks accordingly, you complete the division and solve the problem step-by-step. If you have any more questions, feel free to ask!
