Answer :
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]
