Join the IDNLearn.com community and start finding the answers you need today. Our community is ready to provide in-depth answers and practical solutions to any questions you may have.
Sagot :
To perform the division [tex]\( \left(3x^3 - 5x^2 - 17x - 5\right) \div (3x + 1) \)[/tex], we will apply polynomial long division. Here are the step-by-step details of the process:
1. Set up the division:
We write [tex]\( 3x^3 - 5x^2 - 17x - 5 \)[/tex] as the dividend and [tex]\( 3x + 1 \)[/tex] as the divisor.
2. First step:
Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{3x^3}{3x} = x^2 \][/tex]
Multiply the entire divisor by [tex]\( x^2 \)[/tex] and subtract it from the dividend:
[tex]\[ (3x^3 - 5x^2 - 17x - 5) - (3x^3 + x^2) = -6x^2 - 17x - 5 \][/tex]
3. Second step:
Divide the new leading term of the current dividend by the first term of the divisor:
[tex]\[ \frac{-6x^2}{3x} = -2x \][/tex]
Multiply the entire divisor by [tex]\( -2x \)[/tex] and subtract it from the current dividend:
[tex]\[ (-6x^2 - 17x - 5) - (-6x^2 - 2x) = -15x - 5 \][/tex]
4. Third step:
Divide the new leading term of the current dividend by the first term of the divisor:
[tex]\[ \frac{-15x}{3x} = -5 \][/tex]
Multiply the entire divisor by [tex]\( -5 \)[/tex] and subtract it from the current dividend:
[tex]\[ (-15x - 5) - (-15x - 5) = 0 \][/tex]
Thus, we find that the quotient is [tex]\( x^2 - 2x - 5 \)[/tex] and the remainder is 0.
Therefore, the final answer is:
[tex]\[ x^2 - 2x - 5 + \frac{0}{3x + 1} \][/tex]
Since the remainder is zero, we can simplify this to:
[tex]\[ x^2 - 2x - 5 \][/tex]
1. Set up the division:
We write [tex]\( 3x^3 - 5x^2 - 17x - 5 \)[/tex] as the dividend and [tex]\( 3x + 1 \)[/tex] as the divisor.
2. First step:
Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{3x^3}{3x} = x^2 \][/tex]
Multiply the entire divisor by [tex]\( x^2 \)[/tex] and subtract it from the dividend:
[tex]\[ (3x^3 - 5x^2 - 17x - 5) - (3x^3 + x^2) = -6x^2 - 17x - 5 \][/tex]
3. Second step:
Divide the new leading term of the current dividend by the first term of the divisor:
[tex]\[ \frac{-6x^2}{3x} = -2x \][/tex]
Multiply the entire divisor by [tex]\( -2x \)[/tex] and subtract it from the current dividend:
[tex]\[ (-6x^2 - 17x - 5) - (-6x^2 - 2x) = -15x - 5 \][/tex]
4. Third step:
Divide the new leading term of the current dividend by the first term of the divisor:
[tex]\[ \frac{-15x}{3x} = -5 \][/tex]
Multiply the entire divisor by [tex]\( -5 \)[/tex] and subtract it from the current dividend:
[tex]\[ (-15x - 5) - (-15x - 5) = 0 \][/tex]
Thus, we find that the quotient is [tex]\( x^2 - 2x - 5 \)[/tex] and the remainder is 0.
Therefore, the final answer is:
[tex]\[ x^2 - 2x - 5 + \frac{0}{3x + 1} \][/tex]
Since the remainder is zero, we can simplify this to:
[tex]\[ x^2 - 2x - 5 \][/tex]
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Your questions are important to us at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.