IDNLearn.com provides a comprehensive solution for all your question and answer needs. Get comprehensive answers to all your questions from our network of experienced experts.
Sagot :
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.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.
