Get personalized answers to your specific questions with IDNLearn.com. Get the information you need from our community of experts who provide accurate and comprehensive answers to all your questions.
Sagot :
Sure, let's tackle each part of the problem step-by-step. We'll start with part (a) and then proceed to part (b).
### (a) [tex]\( \mathbf{2p^3 + 4p + 4p^2 + 2 \text{ divided by } 2p^2 + 2 + 2p} \)[/tex]
To divide the polynomial [tex]\( 2p^3 + 4p + 4p^2 + 2 \)[/tex] by [tex]\( 2p^2 + 2 + 2p \)[/tex], we'll follow polynomial long division.
1. Arrange both polynomials in descending order of power:
[tex]\( 2p^3 + 4p^2 + 4p + 2 \)[/tex] (dividend) and [tex]\( 2p^2 + 2p + 2 \)[/tex] (divisor).
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{2p^3}{2p^2} = p \][/tex]
So, the first term of the quotient is [tex]\( p \)[/tex].
3. Multiply the entire divisor by this first term of the quotient:
[tex]\[ p \cdot (2p^2 + 2p + 2) = 2p^3 + 2p^2 + 2p \][/tex]
4. Subtract this product from the original dividend:
[tex]\[ (2p^3 + 4p^2 + 4p + 2) - (2p^3 + 2p^2 + 2p) = 2p^2 + 2p + 2 \][/tex]
5. Now, divide the new dividend by the divisor:
[tex]\[ \frac{2p^2 + 2p + 2}{2p^2 + 2p + 2} = 1 \][/tex]
So, the next term of the quotient is [tex]\( 1 \)[/tex].
6. Multiply the entire divisor by this new term of the quotient:
[tex]\[ 1 \cdot (2p^2 + 2p + 2) = 2p^2 + 2p + 2 \][/tex]
7. Subtract this product from the new dividend:
[tex]\[ (2p^2 + 2p + 2) - (2p^2 + 2p + 2) = 0 \][/tex]
Since the remainder is [tex]\( 0 \)[/tex] and there are no more terms to bring down:
- Quotient: [tex]\( p + 1 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
### (b) [tex]\( \mathbf{24x^3 + 52x - 34x^2 - 30 \text{ divided by } 6x^2 - 4x + 10} \)[/tex]
Now, let's divide [tex]\( 24x^3 + 52x - 34x^2 - 30 \)[/tex] by [tex]\( 6x^2 - 4x + 10 \)[/tex].
1. Arrange both polynomials in descending order of power:
[tex]\( 24x^3 - 34x^2 + 52x - 30 \)[/tex] (dividend) and [tex]\( 6x^2 - 4x + 10 \)[/tex] (divisor).
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{24x^3}{6x^2} = 4x \][/tex]
So, the first term of the quotient is [tex]\( 4x \)[/tex].
3. Multiply the entire divisor by this first term of the quotient:
[tex]\[ 4x \cdot (6x^2 - 4x + 10) = 24x^3 - 16x^2 + 40x \][/tex]
4. Subtract this product from the original dividend:
[tex]\[ (24x^3 - 34x^2 + 52x - 30) - (24x^3 - 16x^2 + 40x) = -18x^2 + 12x - 30 \][/tex]
5. Now, divide the new dividend by the divisor:
[tex]\[ \frac{-18x^2 + 12x - 30}{6x^2 - 4x + 10} = -3 \][/tex]
So, the next term of the quotient is [tex]\( -3 \)[/tex].
6. Multiply the entire divisor by this new term of the quotient:
[tex]\[ -3 \cdot (6x^2 - 4x + 10) = -18x^2 + 12x - 30 \][/tex]
7. Subtract this product from the new dividend:
[tex]\[ (-18x^2 + 12x - 30) - (-18x^2 + 12x - 30) = 0 \][/tex]
Since the remainder is [tex]\( 0 \)[/tex] and there are no more terms to bring down:
- Quotient: [tex]\( 4x - 3 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
### Summary of Results:
- (a)
- Quotient: [tex]\( p + 1 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
- (b)
- Quotient: [tex]\( 4x - 3 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
### (a) [tex]\( \mathbf{2p^3 + 4p + 4p^2 + 2 \text{ divided by } 2p^2 + 2 + 2p} \)[/tex]
To divide the polynomial [tex]\( 2p^3 + 4p + 4p^2 + 2 \)[/tex] by [tex]\( 2p^2 + 2 + 2p \)[/tex], we'll follow polynomial long division.
1. Arrange both polynomials in descending order of power:
[tex]\( 2p^3 + 4p^2 + 4p + 2 \)[/tex] (dividend) and [tex]\( 2p^2 + 2p + 2 \)[/tex] (divisor).
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{2p^3}{2p^2} = p \][/tex]
So, the first term of the quotient is [tex]\( p \)[/tex].
3. Multiply the entire divisor by this first term of the quotient:
[tex]\[ p \cdot (2p^2 + 2p + 2) = 2p^3 + 2p^2 + 2p \][/tex]
4. Subtract this product from the original dividend:
[tex]\[ (2p^3 + 4p^2 + 4p + 2) - (2p^3 + 2p^2 + 2p) = 2p^2 + 2p + 2 \][/tex]
5. Now, divide the new dividend by the divisor:
[tex]\[ \frac{2p^2 + 2p + 2}{2p^2 + 2p + 2} = 1 \][/tex]
So, the next term of the quotient is [tex]\( 1 \)[/tex].
6. Multiply the entire divisor by this new term of the quotient:
[tex]\[ 1 \cdot (2p^2 + 2p + 2) = 2p^2 + 2p + 2 \][/tex]
7. Subtract this product from the new dividend:
[tex]\[ (2p^2 + 2p + 2) - (2p^2 + 2p + 2) = 0 \][/tex]
Since the remainder is [tex]\( 0 \)[/tex] and there are no more terms to bring down:
- Quotient: [tex]\( p + 1 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
### (b) [tex]\( \mathbf{24x^3 + 52x - 34x^2 - 30 \text{ divided by } 6x^2 - 4x + 10} \)[/tex]
Now, let's divide [tex]\( 24x^3 + 52x - 34x^2 - 30 \)[/tex] by [tex]\( 6x^2 - 4x + 10 \)[/tex].
1. Arrange both polynomials in descending order of power:
[tex]\( 24x^3 - 34x^2 + 52x - 30 \)[/tex] (dividend) and [tex]\( 6x^2 - 4x + 10 \)[/tex] (divisor).
2. Divide the first term of the dividend by the first term of the divisor:
[tex]\[ \frac{24x^3}{6x^2} = 4x \][/tex]
So, the first term of the quotient is [tex]\( 4x \)[/tex].
3. Multiply the entire divisor by this first term of the quotient:
[tex]\[ 4x \cdot (6x^2 - 4x + 10) = 24x^3 - 16x^2 + 40x \][/tex]
4. Subtract this product from the original dividend:
[tex]\[ (24x^3 - 34x^2 + 52x - 30) - (24x^3 - 16x^2 + 40x) = -18x^2 + 12x - 30 \][/tex]
5. Now, divide the new dividend by the divisor:
[tex]\[ \frac{-18x^2 + 12x - 30}{6x^2 - 4x + 10} = -3 \][/tex]
So, the next term of the quotient is [tex]\( -3 \)[/tex].
6. Multiply the entire divisor by this new term of the quotient:
[tex]\[ -3 \cdot (6x^2 - 4x + 10) = -18x^2 + 12x - 30 \][/tex]
7. Subtract this product from the new dividend:
[tex]\[ (-18x^2 + 12x - 30) - (-18x^2 + 12x - 30) = 0 \][/tex]
Since the remainder is [tex]\( 0 \)[/tex] and there are no more terms to bring down:
- Quotient: [tex]\( 4x - 3 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
### Summary of Results:
- (a)
- Quotient: [tex]\( p + 1 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
- (b)
- Quotient: [tex]\( 4x - 3 \)[/tex]
- Remainder: [tex]\( 0 \)[/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.
