To find the quotient of [tex]\( 3x^3 - 25x^2 + 12x - 32 \)[/tex] divided by [tex]\( x - 8 \)[/tex] using synthetic division, we will follow these steps:
1. Write down the coefficients of the polynomial [tex]\( 3x^3 - 25x^2 + 12x - 32 \)[/tex]:
[tex]\[
3, -25, 12, -32
\][/tex]
2. Identify the root of the divisor [tex]\( x - 8 \)[/tex], which is [tex]\( 8 \)[/tex].
3. Perform synthetic division:
- Write the root [tex]\( 8 \)[/tex] to the left, and the coefficients to the right:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32
\][/tex]
- Bring down the leading coefficient [tex]\( 3 \)[/tex]:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3
\][/tex]
- Multiply [tex]\( 3 \)[/tex] (the value just brought down) by the root [tex]\( 8 \)[/tex], and write the result below the next coefficient [tex]\(-25\)[/tex]:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3 \\
\quad \quad \quad \quad \downarrow \quad \quad 24
\][/tex]
- Add [tex]\( -25 \)[/tex] and [tex]\( 24 \)[/tex] (the sum is [tex]\(-1\)[/tex]), and write it below:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3 \quad -1 \\
\quad \quad \quad \quad \downarrow \quad 24 \quad \quad \downarrow
\][/tex]
- Multiply [tex]\(-1\)[/tex] (the value just written down) by the root [tex]\( 8 \)[/tex], and write the result below the next coefficient [tex]\( 12 \)[/tex]:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3 \quad -1 \\
\quad \quad \quad \quad \downarrow \quad 24 \quad \quad \downarrow \quad -8
\][/tex]
- Add [tex]\( 12 \)[/tex] and [tex]\(-8\)[/tex] (the sum is [tex]\( 4 \)[/tex]), and write it below:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3 \quad -1 \quad 4 \\
\quad \quad \quad \quad \downarrow \quad 24 \quad \quad \downarrow \quad -8 \quad \quad \downarrow
\][/tex]
- Multiply [tex]\( 4 \)[/tex] (the value just written down) by the root [tex]\( 8 \)[/tex], and write the result below the next coefficient [tex]\(-32\)[/tex]:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3 \quad -1 \quad 4 \\
\quad \quad \quad \quad \downarrow \quad 24 \quad \quad \downarrow \quad -8 \quad \quad \downarrow \quad 32
\][/tex]
- Add [tex]\(-32\)[/tex] and [tex]\( 32 \)[/tex] (the sum is [tex]\( 0 \)[/tex]), and write it below as the remainder:
[tex]\[
8 \quad | \quad 3 \quad -25 \quad 12 \quad -32 \\
\quad \quad \quad \quad 3 \quad -1 \quad 4 \quad 0 \\
\quad \quad \quad \quad \downarrow \quad 24 \quad \quad \downarrow \quad -8 \quad \downarrow \quad 32
\][/tex]
So, the quotient is [tex]\( 3x^2 - x + 4 \)[/tex] and the remainder is [tex]\( 0 \)[/tex].
