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Sagot :
Certainly! Let's go through each part step-by-step to solve the polynomial divisions.
### (a) [tex]\((10 + 2x - 4x^2) \div (2x + 2)\)[/tex]
First, we arrange the polynomials in descending powers of [tex]\( x \)[/tex]:
Dividend: [tex]\(-4x^2 + 2x + 10\)[/tex]
Divisor: [tex]\(2x + 2\)[/tex]
1. First Term:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{-4x^2}{2x} = -2x \][/tex]
Multiply [tex]\(-2x\)[/tex] by the divisor:
[tex]\[ -2x(2x + 2) = -4x^2 - 4x \][/tex]
Subtract from the dividend:
[tex]\[ (-4x^2 + 2x + 10) - (-4x^2 - 4x) = 6x + 10 \][/tex]
2. Second Term:
Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{6x}{2x} = 3 \][/tex]
Multiply [tex]\(3\)[/tex] by the divisor:
[tex]\[ 3(2x + 2) = 6x + 6 \][/tex]
Subtract again:
[tex]\[ (6x + 10) - (6x + 6) = 4 \][/tex]
The quotient is [tex]\(-2x + 3\)[/tex] and the remainder is [tex]\(4\)[/tex]:
[tex]\[ \boxed{-2x + 3} \text{ (remainder } 4\text{)} \][/tex]
### (b) [tex]\((5m^3 + 35 - 30m^2) \div (5m - 5)\)[/tex]
Arrange the polynomials in descending powers of [tex]\( m \)[/tex]:
Dividend: [tex]\(5m^3 - 30m^2 + 35\)[/tex]
Divisor: [tex]\(5m - 5\)[/tex]
1. First Term:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{5m^3}{5m} = m^2 \][/tex]
Multiply [tex]\(m^2\)[/tex] by the divisor:
[tex]\[ m^2(5m - 5) = 5m^3 - 5m^2 \][/tex]
Subtract from the dividend:
[tex]\[ (5m^3 - 30m^2 + 35) - (5m^3 - 5m^2) = -25m^2 + 35 \][/tex]
2. Second Term:
Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{-25m^2}{5m} = -5m \][/tex]
Multiply [tex]\(-5m\)[/tex] by the divisor:
[tex]\[ -5m(5m - 5) = -25m^2 + 25m \][/tex]
Subtract again:
[tex]\[ (-25m^2 + 35) - (-25m^2 + 25m) = -25m + 35 \][/tex]
3. Third Term:
Divide the new leading term:
[tex]\[ \frac{-25m}{5m} = -5 \][/tex]
Multiply [tex]\(-5\)[/tex] by the divisor:
[tex]\[ -5(5m - 5) = -25m + 25 \][/tex]
Subtract again:
[tex]\[ (-25m + 35) - (-25m + 25) = 10 \][/tex]
The quotient is [tex]\(m^2 - 5m - 5\)[/tex] and the remainder is [tex]\(10\)[/tex]:
[tex]\[ \boxed{m^2 - 5m - 5} \text{ (remainder } 10\text{)} \][/tex]
### (c) [tex]\((1 + p^3) \div (p + 1)\)[/tex]
1. First Term:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{p^3}{p} = p^2 \][/tex]
Multiply [tex]\(p^2\)[/tex] by the divisor:
[tex]\[ p^2(p + 1) = p^3 + p^2 \][/tex]
Subtract from the dividend:
[tex]\[ (p^3 + 1) - (p^3 + p^2) = -p^2 + 1 \][/tex]
2. Second Term:
Divide the new leading term:
[tex]\[ \frac{-p^2}{p} = -p \][/tex]
Multiply [tex]\(-p\)[/tex] by the divisor:
[tex]\[ -p(p + 1) = -p^2 - p \][/tex]
Subtract again:
[tex]\[ (-p^2 + 1) - (-p^2 - p) = p + 1 \][/tex]
3. Third Term:
Divide the new leading term:
[tex]\[ \frac{p}{p} = 1 \][/tex]
Multiply [tex]\(1\)[/tex] by the divisor:
[tex]\[ 1(p + 1) = p + 1 \][/tex]
Subtract again:
[tex]\[ (p + 1) - (p + 1) = 0 \][/tex]
The quotient is [tex]\(p^2 - p + 1\)[/tex] and the remainder is [tex]\(0\)[/tex]:
[tex]\[ \boxed{p^2 - p + 1} \][/tex]
### (d) [tex]\((8p^2 + 28p - 36) \div (4p - 24)\)[/tex]
First, factor out any common terms in the divisor:
Divisor: [tex]\(4(p - 6)\)[/tex]
Dividend: [tex]\(8p^2 + 28p - 36\)[/tex]
Rewrite the divisor:
[tex]\[ (8p^2 + 28p - 36) \div 4(p - 6) \][/tex]
Divide each term in the dividend by 4:
[tex]\[ 2p^2 + 7p - 9 \text{ divided by } (p - 6) \][/tex]
1. First Term:
Divide the leading term:
[tex]\[ \frac{2p^2}{p} = 2p \][/tex]
Multiply [tex]\(2p\)[/tex] by the divisor:
[tex]\[ 2p(p - 6) = 2p^2 - 12p \][/tex]
Subtract from the dividend:
[tex]\[ (2p^2 + 7p - 9) - (2p^2 - 12p) = 19p - 9 \][/tex]
2. Second Term:
Divide the new leading term:
[tex]\[ \frac{19p}{p} = 19 \][/tex]
Multiply [tex]\(19\)[/tex] by the divisor:
[tex]\[ 19(p - 6) = 19p - 114 \][/tex]
Subtract again:
[tex]\[ (19p - 9) - (19p - 114) = 105 \][/tex]
The quotient is [tex]\(2p + 19\)[/tex] and the remainder is [tex]\(105\)[/tex]:
[tex]\[ \boxed{2p + 19} \text{ (remainder } 105\text{)} \][/tex]
### (a) [tex]\((10 + 2x - 4x^2) \div (2x + 2)\)[/tex]
First, we arrange the polynomials in descending powers of [tex]\( x \)[/tex]:
Dividend: [tex]\(-4x^2 + 2x + 10\)[/tex]
Divisor: [tex]\(2x + 2\)[/tex]
1. First Term:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{-4x^2}{2x} = -2x \][/tex]
Multiply [tex]\(-2x\)[/tex] by the divisor:
[tex]\[ -2x(2x + 2) = -4x^2 - 4x \][/tex]
Subtract from the dividend:
[tex]\[ (-4x^2 + 2x + 10) - (-4x^2 - 4x) = 6x + 10 \][/tex]
2. Second Term:
Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{6x}{2x} = 3 \][/tex]
Multiply [tex]\(3\)[/tex] by the divisor:
[tex]\[ 3(2x + 2) = 6x + 6 \][/tex]
Subtract again:
[tex]\[ (6x + 10) - (6x + 6) = 4 \][/tex]
The quotient is [tex]\(-2x + 3\)[/tex] and the remainder is [tex]\(4\)[/tex]:
[tex]\[ \boxed{-2x + 3} \text{ (remainder } 4\text{)} \][/tex]
### (b) [tex]\((5m^3 + 35 - 30m^2) \div (5m - 5)\)[/tex]
Arrange the polynomials in descending powers of [tex]\( m \)[/tex]:
Dividend: [tex]\(5m^3 - 30m^2 + 35\)[/tex]
Divisor: [tex]\(5m - 5\)[/tex]
1. First Term:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{5m^3}{5m} = m^2 \][/tex]
Multiply [tex]\(m^2\)[/tex] by the divisor:
[tex]\[ m^2(5m - 5) = 5m^3 - 5m^2 \][/tex]
Subtract from the dividend:
[tex]\[ (5m^3 - 30m^2 + 35) - (5m^3 - 5m^2) = -25m^2 + 35 \][/tex]
2. Second Term:
Divide the new leading term by the leading term of the divisor:
[tex]\[ \frac{-25m^2}{5m} = -5m \][/tex]
Multiply [tex]\(-5m\)[/tex] by the divisor:
[tex]\[ -5m(5m - 5) = -25m^2 + 25m \][/tex]
Subtract again:
[tex]\[ (-25m^2 + 35) - (-25m^2 + 25m) = -25m + 35 \][/tex]
3. Third Term:
Divide the new leading term:
[tex]\[ \frac{-25m}{5m} = -5 \][/tex]
Multiply [tex]\(-5\)[/tex] by the divisor:
[tex]\[ -5(5m - 5) = -25m + 25 \][/tex]
Subtract again:
[tex]\[ (-25m + 35) - (-25m + 25) = 10 \][/tex]
The quotient is [tex]\(m^2 - 5m - 5\)[/tex] and the remainder is [tex]\(10\)[/tex]:
[tex]\[ \boxed{m^2 - 5m - 5} \text{ (remainder } 10\text{)} \][/tex]
### (c) [tex]\((1 + p^3) \div (p + 1)\)[/tex]
1. First Term:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[ \frac{p^3}{p} = p^2 \][/tex]
Multiply [tex]\(p^2\)[/tex] by the divisor:
[tex]\[ p^2(p + 1) = p^3 + p^2 \][/tex]
Subtract from the dividend:
[tex]\[ (p^3 + 1) - (p^3 + p^2) = -p^2 + 1 \][/tex]
2. Second Term:
Divide the new leading term:
[tex]\[ \frac{-p^2}{p} = -p \][/tex]
Multiply [tex]\(-p\)[/tex] by the divisor:
[tex]\[ -p(p + 1) = -p^2 - p \][/tex]
Subtract again:
[tex]\[ (-p^2 + 1) - (-p^2 - p) = p + 1 \][/tex]
3. Third Term:
Divide the new leading term:
[tex]\[ \frac{p}{p} = 1 \][/tex]
Multiply [tex]\(1\)[/tex] by the divisor:
[tex]\[ 1(p + 1) = p + 1 \][/tex]
Subtract again:
[tex]\[ (p + 1) - (p + 1) = 0 \][/tex]
The quotient is [tex]\(p^2 - p + 1\)[/tex] and the remainder is [tex]\(0\)[/tex]:
[tex]\[ \boxed{p^2 - p + 1} \][/tex]
### (d) [tex]\((8p^2 + 28p - 36) \div (4p - 24)\)[/tex]
First, factor out any common terms in the divisor:
Divisor: [tex]\(4(p - 6)\)[/tex]
Dividend: [tex]\(8p^2 + 28p - 36\)[/tex]
Rewrite the divisor:
[tex]\[ (8p^2 + 28p - 36) \div 4(p - 6) \][/tex]
Divide each term in the dividend by 4:
[tex]\[ 2p^2 + 7p - 9 \text{ divided by } (p - 6) \][/tex]
1. First Term:
Divide the leading term:
[tex]\[ \frac{2p^2}{p} = 2p \][/tex]
Multiply [tex]\(2p\)[/tex] by the divisor:
[tex]\[ 2p(p - 6) = 2p^2 - 12p \][/tex]
Subtract from the dividend:
[tex]\[ (2p^2 + 7p - 9) - (2p^2 - 12p) = 19p - 9 \][/tex]
2. Second Term:
Divide the new leading term:
[tex]\[ \frac{19p}{p} = 19 \][/tex]
Multiply [tex]\(19\)[/tex] by the divisor:
[tex]\[ 19(p - 6) = 19p - 114 \][/tex]
Subtract again:
[tex]\[ (19p - 9) - (19p - 114) = 105 \][/tex]
The quotient is [tex]\(2p + 19\)[/tex] and the remainder is [tex]\(105\)[/tex]:
[tex]\[ \boxed{2p + 19} \text{ (remainder } 105\text{)} \][/tex]
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