Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Experience the ease of finding accurate answers to your questions from a knowledgeable community of professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Let's perform the polynomial division step-by-step to find the quotient and the remainder of [tex]\( \frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} \)[/tex].
1. Setup the division:
We want to divide the polynomial [tex]\( 10x^4 - 14x^3 - 10x^2 + 6x - 10 \)[/tex] (the numerator) by [tex]\( x^3 - 3x^2 + x - 2 \)[/tex] (the denominator).
2. First term of the quotient:
[tex]\[ \frac{10x^4}{x^3} = 10x \][/tex]
Multiply [tex]\( 10x \)[/tex] by the whole denominator:
[tex]\[ 10x \cdot (x^3 - 3x^2 + x - 2) = 10x^4 - 30x^3 + 10x^2 - 20x \][/tex]
Subtract this result from the original numerator:
[tex]\[ (10x^4 - 14x^3 - 10x^2 + 6x - 10) - (10x^4 - 30x^3 + 10x^2 - 20x) \][/tex]
Simplify:
[tex]\[ 16x^3 - 20x^2 + 26x - 10 \][/tex]
3. Second term of the quotient:
[tex]\[ \frac{16x^3}{x^3} = 16 \][/tex]
Multiply [tex]\( 16 \)[/tex] by the whole denominator:
[tex]\[ 16 \cdot (x^3 - 3x^2 + x - 2) = 16x^3 - 48x^2 + 16x - 32 \][/tex]
Subtract this result from the remainder:
[tex]\[ (16x^3 - 20x^2 + 26x - 10) - (16x^3 - 48x^2 + 16x - 32) \][/tex]
Simplify:
[tex]\[ 28x^2 + 10x + 22 \][/tex]
Now, the quotient of the division is [tex]\( 10x + 16 \)[/tex], and the remainder is [tex]\( 28x^2 + 10x + 22 \)[/tex].
Thus, putting it all together:
[tex]\[ \frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} = 10x + 16 + \frac{28x^2 + 10x + 22}{x^3 - 3x^2 + x - 2} \][/tex]
Finally, filling in the blanks:
- The quotient is [tex]\( \boxed{10} \)[/tex] [tex]\( x + \)[/tex] [tex]\(\boxed{16}\)[/tex]
- The remainder is [tex]\(\boxed{28}\)[/tex] [tex]\(x^2 +\)[/tex] [tex]\(\boxed{10}\)[/tex] [tex]\(x +\)[/tex] [tex]\(\boxed{22}\)[/tex]
1. Setup the division:
We want to divide the polynomial [tex]\( 10x^4 - 14x^3 - 10x^2 + 6x - 10 \)[/tex] (the numerator) by [tex]\( x^3 - 3x^2 + x - 2 \)[/tex] (the denominator).
2. First term of the quotient:
[tex]\[ \frac{10x^4}{x^3} = 10x \][/tex]
Multiply [tex]\( 10x \)[/tex] by the whole denominator:
[tex]\[ 10x \cdot (x^3 - 3x^2 + x - 2) = 10x^4 - 30x^3 + 10x^2 - 20x \][/tex]
Subtract this result from the original numerator:
[tex]\[ (10x^4 - 14x^3 - 10x^2 + 6x - 10) - (10x^4 - 30x^3 + 10x^2 - 20x) \][/tex]
Simplify:
[tex]\[ 16x^3 - 20x^2 + 26x - 10 \][/tex]
3. Second term of the quotient:
[tex]\[ \frac{16x^3}{x^3} = 16 \][/tex]
Multiply [tex]\( 16 \)[/tex] by the whole denominator:
[tex]\[ 16 \cdot (x^3 - 3x^2 + x - 2) = 16x^3 - 48x^2 + 16x - 32 \][/tex]
Subtract this result from the remainder:
[tex]\[ (16x^3 - 20x^2 + 26x - 10) - (16x^3 - 48x^2 + 16x - 32) \][/tex]
Simplify:
[tex]\[ 28x^2 + 10x + 22 \][/tex]
Now, the quotient of the division is [tex]\( 10x + 16 \)[/tex], and the remainder is [tex]\( 28x^2 + 10x + 22 \)[/tex].
Thus, putting it all together:
[tex]\[ \frac{10x^4 - 14x^3 - 10x^2 + 6x - 10}{x^3 - 3x^2 + x - 2} = 10x + 16 + \frac{28x^2 + 10x + 22}{x^3 - 3x^2 + x - 2} \][/tex]
Finally, filling in the blanks:
- The quotient is [tex]\( \boxed{10} \)[/tex] [tex]\( x + \)[/tex] [tex]\(\boxed{16}\)[/tex]
- The remainder is [tex]\(\boxed{28}\)[/tex] [tex]\(x^2 +\)[/tex] [tex]\(\boxed{10}\)[/tex] [tex]\(x +\)[/tex] [tex]\(\boxed{22}\)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.