Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Our platform provides a seamless experience for finding reliable answers from a knowledgeable network of professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
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].
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].
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
