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Let's address each part of the question step-by-step and verify the answers:
### (a) Divide [tex]\( 10 + 2x - 4x^2 \)[/tex] by [tex]\( 2x + 2 \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{10 + 2x - 4x^2}{2x + 2} \][/tex]
2. Rearrange the numerator in the standard form [tex]\( -4x^2 + 2x + 10 \)[/tex].
3. Perform polynomial long division:
1. First term: [tex]\((-4x^2 \div 2x) = -2x\)[/tex]
2. Multiply [tex]\((-2x) \times (2x + 2) = -4x^2 - 4x\)[/tex]
3. Subtract: [tex]\((-4x^2 + 2x + 10) - (-4x^2 - 4x) = 6x + 10\)[/tex]
4. Next term: [tex]\((6x \div 2x) = 3\)[/tex]
5. Multiply [tex]\( 3 \times (2x + 2) = 6x + 6\)[/tex]
6. Subtract: [tex]\(6x + 10 - (6x + 6) = 4\)[/tex]
4. So, the quotient is [tex]\(3 - 2x\)[/tex] and the remainder is [tex]\(4\)[/tex].
Answer:
[tex]\[ \frac{10 + 2x - 4x^2}{2x + 2} = -2x + 3 \text{ with remainder } 4 \][/tex]
### (b) Divide [tex]\( 5m^2 + 39 \)[/tex] by [tex]\( 9m \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{5m^2 + 39}{9m} \][/tex]
2. Perform polynomial long division:
1. First term: [tex]\((5m^2 \div 9m) = \frac{5m}{9}\)[/tex]
2. Multiply: [tex]\(\frac{5m}{9} \times 9m = 5m^2\)[/tex]
3. Subtract: [tex]\(5m^2 + 39 - 5m^2 = 39\)[/tex]
3. So, the quotient is [tex]\(\frac{5m}{9}\)[/tex] and the remainder is [tex]\(39\)[/tex].
Answer:
[tex]\[ \frac{5m^2 + 39}{9m} = \frac{5m}{9} \text{ with remainder } 39 \][/tex]
### (c) Divide [tex]\( 1 + p^3 \)[/tex] by [tex]\( p + 1 \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{1 + p^3}{p + 1} \][/tex]
2. Perform polynomial long division:
1. First term: [tex]\((p^3 \div p) = p^2\)[/tex]
2. Multiply: [tex]\( p^2 \times (p + 1) = p^3 + p^2\)[/tex]
3. Subtract: [tex]\(1 + p^3 - (p^3 + p^2) = 1 - p^2\)[/tex]
4. Second term: [tex]\(( - p^2 \div p) = -p\)[/tex]
5. Multiply: [tex]\((-p) \times (p + 1) = -p^2 - p\)[/tex]
6. Subtract: [tex]\(1 - p^2 - (-p^2 - p) = 1 + p\)[/tex]
7. Third term: [tex]\((p \div p) = 1\)[/tex]
8. Multiply: [tex]\(1 \times (p + 1) = p + 1\)[/tex]
9. Subtract: [tex]\(1 + p - (p + 1) = 0\)[/tex]
3. So, the quotient is [tex]\( p^2 - p + 1 \)[/tex] and the remainder is [tex]\(0\)[/tex].
Answer:
[tex]\[ \frac{1 + p^3}{p + 1} = p^2 - p + 1 \text{ with remainder } 0 \][/tex]
### (d) Divide [tex]\( 8p^2 + 28p - 368 \)[/tex] by [tex]\( 4p + 24 \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{8p^2 + 28p - 368}{4p + 24} \][/tex]
2. Perform polynomial long division:
1. First term: [tex]\((8p^2 \div 4p) = 2p\)[/tex]
2. Multiply: [tex]\( 2p \times (4p + 24) = 8p^2 + 48p\)[/tex]
3. Subtract: [tex]\(8p^2 + 28p - 368 - (8p^2 + 48p) = -20p - 368\)[/tex]
4. Second term: [tex]\((-20p \div 4p) = -5\)[/tex]
5. Multiply: [tex]\((-5) \times (4p + 24) = -20p - 120\)[/tex]
6. Subtract: [tex]\(-20p - 368 - (-20p - 120) = -248\)[/tex]
3. So, the quotient is [tex]\(2p - 5\)[/tex] and the remainder is [tex]\(-248\)[/tex].
Answer:
[tex]\[ \frac{8p^2 + 28p - 368}{4p + 24} = 2p - 5 \text{ with remainder } -248 \][/tex]
These are the step-by-step solutions and verifications for each problem.
### (a) Divide [tex]\( 10 + 2x - 4x^2 \)[/tex] by [tex]\( 2x + 2 \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{10 + 2x - 4x^2}{2x + 2} \][/tex]
2. Rearrange the numerator in the standard form [tex]\( -4x^2 + 2x + 10 \)[/tex].
3. Perform polynomial long division:
1. First term: [tex]\((-4x^2 \div 2x) = -2x\)[/tex]
2. Multiply [tex]\((-2x) \times (2x + 2) = -4x^2 - 4x\)[/tex]
3. Subtract: [tex]\((-4x^2 + 2x + 10) - (-4x^2 - 4x) = 6x + 10\)[/tex]
4. Next term: [tex]\((6x \div 2x) = 3\)[/tex]
5. Multiply [tex]\( 3 \times (2x + 2) = 6x + 6\)[/tex]
6. Subtract: [tex]\(6x + 10 - (6x + 6) = 4\)[/tex]
4. So, the quotient is [tex]\(3 - 2x\)[/tex] and the remainder is [tex]\(4\)[/tex].
Answer:
[tex]\[ \frac{10 + 2x - 4x^2}{2x + 2} = -2x + 3 \text{ with remainder } 4 \][/tex]
### (b) Divide [tex]\( 5m^2 + 39 \)[/tex] by [tex]\( 9m \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{5m^2 + 39}{9m} \][/tex]
2. Perform polynomial long division:
1. First term: [tex]\((5m^2 \div 9m) = \frac{5m}{9}\)[/tex]
2. Multiply: [tex]\(\frac{5m}{9} \times 9m = 5m^2\)[/tex]
3. Subtract: [tex]\(5m^2 + 39 - 5m^2 = 39\)[/tex]
3. So, the quotient is [tex]\(\frac{5m}{9}\)[/tex] and the remainder is [tex]\(39\)[/tex].
Answer:
[tex]\[ \frac{5m^2 + 39}{9m} = \frac{5m}{9} \text{ with remainder } 39 \][/tex]
### (c) Divide [tex]\( 1 + p^3 \)[/tex] by [tex]\( p + 1 \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{1 + p^3}{p + 1} \][/tex]
2. Perform polynomial long division:
1. First term: [tex]\((p^3 \div p) = p^2\)[/tex]
2. Multiply: [tex]\( p^2 \times (p + 1) = p^3 + p^2\)[/tex]
3. Subtract: [tex]\(1 + p^3 - (p^3 + p^2) = 1 - p^2\)[/tex]
4. Second term: [tex]\(( - p^2 \div p) = -p\)[/tex]
5. Multiply: [tex]\((-p) \times (p + 1) = -p^2 - p\)[/tex]
6. Subtract: [tex]\(1 - p^2 - (-p^2 - p) = 1 + p\)[/tex]
7. Third term: [tex]\((p \div p) = 1\)[/tex]
8. Multiply: [tex]\(1 \times (p + 1) = p + 1\)[/tex]
9. Subtract: [tex]\(1 + p - (p + 1) = 0\)[/tex]
3. So, the quotient is [tex]\( p^2 - p + 1 \)[/tex] and the remainder is [tex]\(0\)[/tex].
Answer:
[tex]\[ \frac{1 + p^3}{p + 1} = p^2 - p + 1 \text{ with remainder } 0 \][/tex]
### (d) Divide [tex]\( 8p^2 + 28p - 368 \)[/tex] by [tex]\( 4p + 24 \)[/tex]
1. Write the division in polynomial form:
[tex]\[ \frac{8p^2 + 28p - 368}{4p + 24} \][/tex]
2. Perform polynomial long division:
1. First term: [tex]\((8p^2 \div 4p) = 2p\)[/tex]
2. Multiply: [tex]\( 2p \times (4p + 24) = 8p^2 + 48p\)[/tex]
3. Subtract: [tex]\(8p^2 + 28p - 368 - (8p^2 + 48p) = -20p - 368\)[/tex]
4. Second term: [tex]\((-20p \div 4p) = -5\)[/tex]
5. Multiply: [tex]\((-5) \times (4p + 24) = -20p - 120\)[/tex]
6. Subtract: [tex]\(-20p - 368 - (-20p - 120) = -248\)[/tex]
3. So, the quotient is [tex]\(2p - 5\)[/tex] and the remainder is [tex]\(-248\)[/tex].
Answer:
[tex]\[ \frac{8p^2 + 28p - 368}{4p + 24} = 2p - 5 \text{ with remainder } -248 \][/tex]
These are the step-by-step solutions and verifications for each problem.
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