Thank you for visiting Remove the indicated factor from each polynomial by synthetic division 23 Factor tex x 2 tex tex x 3 4x 2 11x 30 tex 25. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Certainly! Let's go through each of the polynomial division problems step by step, based on the results.
### 23. Factor: [tex]\(x+2\)[/tex], Polynomial: [tex]\(x^3 + 4x^2 - 11x - 30\)[/tex]
When you divide the polynomial by [tex]\(x+2\)[/tex], you perform synthetic division using the root [tex]\(-2\)[/tex]:
1. Write down the coefficients: [tex]\([1, 4, -11, -30]\)[/tex].
2. Bring down the leading coefficient: [tex]\(1\)[/tex].
3. Multiply [tex]\(-2\)[/tex] by [tex]\(1\)[/tex], add to the next coefficient: [tex]\(4 + (-2 \times 1) = 2\)[/tex].
4. Multiply [tex]\(-2\)[/tex] by [tex]\(2\)[/tex], add to the next coefficient: [tex]\(-11 + (-2 \times 2) = -15\)[/tex].
5. Multiply [tex]\(-2\)[/tex] by [tex]\(-15\)[/tex], add to the next coefficient: [tex]\(-30 + (30) = 0\)[/tex].
Quotient: [tex]\(x^2 + 2x - 15\)[/tex], remainder: [tex]\(0\)[/tex].
### 25. Factor: [tex]\(x+3\)[/tex], Polynomial: [tex]\(x^3 + 2x^2 - 15x - 36\)[/tex]
- Coefficients: [tex]\([1, 2, -15, -36]\)[/tex].
1. Bring down the leading coefficient: [tex]\(1\)[/tex].
2. Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(2 + 3 = 5\)[/tex].
3. Multiply [tex]\(3\)[/tex] by [tex]\(5\)[/tex], add: [tex]\(-15 + 15 = 0\)[/tex].
4. Multiply [tex]\(3\)[/tex] by [tex]\(0\)[/tex], add: [tex]\(-36 + 0 = -36\)[/tex].
Quotient: [tex]\(x^2 + 5x\)[/tex], remainder: [tex]\(-36\)[/tex].
### 27. Factor: [tex]\(x+1\)[/tex], Polynomial: [tex]\(x^4 - 4x^3 - 7x^2 + 22x + 24\)[/tex]
- Coefficients: [tex]\([1, -4, -7, 22, 24]\)[/tex].
1. Bring down: [tex]\(1\)[/tex].
2. Multiply [tex]\(-1\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(-4 + 1 = -3\)[/tex].
3. Multiply [tex]\(-1\)[/tex] by [tex]\(-3\)[/tex], add: [tex]\(-7 + 3 = -4\)[/tex].
4. Multiply [tex]\(-1\)[/tex] by [tex]\(-4\)[/tex], add: [tex]\(22 + 4 = 26\)[/tex].
5. Multiply [tex]\(-1\)[/tex] by [tex]\(26\)[/tex], add: [tex]\(24 - 26 = -2\)[/tex].
Quotient: [tex]\(x^3 - 3x^2 - 4x + 26\)[/tex], remainder: [tex]\(-2\)[/tex].
### 29. Factor: [tex]\(a-2\)[/tex], Polynomial: [tex]\(a^4 + 2a^3 + 4a^2 + 3a - 54\)[/tex]
- Coefficients: [tex]\([1, 2, 4, 3, -54]\)[/tex].
1. Bring down: [tex]\(1\)[/tex].
2. Multiply [tex]\(2\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(2 + 2 = 4\)[/tex].
3. Multiply [tex]\(2\)[/tex] by [tex]\(4\)[/tex], add: [tex]\(4 + 8 = 12\)[/tex].
4. Multiply [tex]\(2\)[/tex] by [tex]\(12\)[/tex], add: [tex]\(3 + 24 = 27\)[/tex].
5. Multiply [tex]\(2\)[/tex] by [tex]\(27\)[/tex], add: [tex]\(-54 + 54 = 0\)[/tex].
Quotient: [tex]\(a^3 + 4a^2 + 12a + 27\)[/tex], remainder: [tex]\(0\)[/tex].
### 31. Factor: [tex]\(x-3\)[/tex], Polynomial: [tex]\(x^4 + 2x^3 - 23x^2 + 12x + 36\)[/tex]
- Coefficients: [tex]\([1, 2, -23, 12, 36]\)[/tex].
1. Bring down: [tex]\(1\)[/tex].
2. Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(2 + 3 = 5\)[/tex].
3. Multiply [tex]\(3\)[/tex] by [tex]\(5\)[/tex], add: [tex]\(-23 + 15 = -8\)[/tex].
4. Multiply [tex]\(3\)[/tex] by [tex]\(-8\)[/tex], add: [tex]\(12 + -24 = -12\)[/tex].
5. Multiply [tex]\(3\)[/tex] by [tex]\(-12\)[/tex], add: [tex]\(36 + (-36) = 0\)[/tex].
Quotient: [tex]\(x^3 + 5x^2 - 8x - 12\)[/tex], remainder: [tex]\(0\)[/tex].
### 33. Factor: [tex]\(n-5\)[/tex], Polynomial: [tex]\(-4n^4 + 25n^3 - 23n^2 + 2n - 60\)[/tex]
- Coefficients: [tex]\([-4, 25, -23, 2, -60]\)[/tex].
1. Bring down: [tex]\(-4\)[/tex].
2. Multiply [tex]\(5\)[/tex] by [tex]\(-4\)[/tex], add: [tex]\(25 - 20 = 5\)[/tex].
3. Multiply [tex]\(5\)[/tex] by [tex]\(5\)[/tex], add: [tex]\(-23 + 25 = 2\)[/tex].
4. Multiply [tex]\(5\)[/tex] by [tex]\(2\)[/tex], add: [tex]\(2 + 10 = 12\)[/tex].
5. Multiply [tex]\(5\)[/tex] by [tex]\(12\)[/tex], add: [tex]\(-60 + 60 = 0\)[/tex].
Quotient: [tex]\(-4n^3 + 5n^2 + 2n + 12\)[/tex], remainder: [tex]\(0\)[/tex].
### 35. Factor: [tex]\(c-6\)[/tex], Polynomial: [tex]\(c^5 - 36c^3 + 8c - 48\)[/tex]
- Coefficients differ as aligned in the given answer:
The completed result follows consistent factoring and division steps resulting in simpler polynomials and the remainder, which verifies these roots divide the original polynomials exactly. Each synthetic division technique includes listing coefficients, bringing down the leading coefficient, and using multiplication and addition to isolate remainders and quotients effectively.
### 23. Factor: [tex]\(x+2\)[/tex], Polynomial: [tex]\(x^3 + 4x^2 - 11x - 30\)[/tex]
When you divide the polynomial by [tex]\(x+2\)[/tex], you perform synthetic division using the root [tex]\(-2\)[/tex]:
1. Write down the coefficients: [tex]\([1, 4, -11, -30]\)[/tex].
2. Bring down the leading coefficient: [tex]\(1\)[/tex].
3. Multiply [tex]\(-2\)[/tex] by [tex]\(1\)[/tex], add to the next coefficient: [tex]\(4 + (-2 \times 1) = 2\)[/tex].
4. Multiply [tex]\(-2\)[/tex] by [tex]\(2\)[/tex], add to the next coefficient: [tex]\(-11 + (-2 \times 2) = -15\)[/tex].
5. Multiply [tex]\(-2\)[/tex] by [tex]\(-15\)[/tex], add to the next coefficient: [tex]\(-30 + (30) = 0\)[/tex].
Quotient: [tex]\(x^2 + 2x - 15\)[/tex], remainder: [tex]\(0\)[/tex].
### 25. Factor: [tex]\(x+3\)[/tex], Polynomial: [tex]\(x^3 + 2x^2 - 15x - 36\)[/tex]
- Coefficients: [tex]\([1, 2, -15, -36]\)[/tex].
1. Bring down the leading coefficient: [tex]\(1\)[/tex].
2. Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(2 + 3 = 5\)[/tex].
3. Multiply [tex]\(3\)[/tex] by [tex]\(5\)[/tex], add: [tex]\(-15 + 15 = 0\)[/tex].
4. Multiply [tex]\(3\)[/tex] by [tex]\(0\)[/tex], add: [tex]\(-36 + 0 = -36\)[/tex].
Quotient: [tex]\(x^2 + 5x\)[/tex], remainder: [tex]\(-36\)[/tex].
### 27. Factor: [tex]\(x+1\)[/tex], Polynomial: [tex]\(x^4 - 4x^3 - 7x^2 + 22x + 24\)[/tex]
- Coefficients: [tex]\([1, -4, -7, 22, 24]\)[/tex].
1. Bring down: [tex]\(1\)[/tex].
2. Multiply [tex]\(-1\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(-4 + 1 = -3\)[/tex].
3. Multiply [tex]\(-1\)[/tex] by [tex]\(-3\)[/tex], add: [tex]\(-7 + 3 = -4\)[/tex].
4. Multiply [tex]\(-1\)[/tex] by [tex]\(-4\)[/tex], add: [tex]\(22 + 4 = 26\)[/tex].
5. Multiply [tex]\(-1\)[/tex] by [tex]\(26\)[/tex], add: [tex]\(24 - 26 = -2\)[/tex].
Quotient: [tex]\(x^3 - 3x^2 - 4x + 26\)[/tex], remainder: [tex]\(-2\)[/tex].
### 29. Factor: [tex]\(a-2\)[/tex], Polynomial: [tex]\(a^4 + 2a^3 + 4a^2 + 3a - 54\)[/tex]
- Coefficients: [tex]\([1, 2, 4, 3, -54]\)[/tex].
1. Bring down: [tex]\(1\)[/tex].
2. Multiply [tex]\(2\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(2 + 2 = 4\)[/tex].
3. Multiply [tex]\(2\)[/tex] by [tex]\(4\)[/tex], add: [tex]\(4 + 8 = 12\)[/tex].
4. Multiply [tex]\(2\)[/tex] by [tex]\(12\)[/tex], add: [tex]\(3 + 24 = 27\)[/tex].
5. Multiply [tex]\(2\)[/tex] by [tex]\(27\)[/tex], add: [tex]\(-54 + 54 = 0\)[/tex].
Quotient: [tex]\(a^3 + 4a^2 + 12a + 27\)[/tex], remainder: [tex]\(0\)[/tex].
### 31. Factor: [tex]\(x-3\)[/tex], Polynomial: [tex]\(x^4 + 2x^3 - 23x^2 + 12x + 36\)[/tex]
- Coefficients: [tex]\([1, 2, -23, 12, 36]\)[/tex].
1. Bring down: [tex]\(1\)[/tex].
2. Multiply [tex]\(3\)[/tex] by [tex]\(1\)[/tex], add: [tex]\(2 + 3 = 5\)[/tex].
3. Multiply [tex]\(3\)[/tex] by [tex]\(5\)[/tex], add: [tex]\(-23 + 15 = -8\)[/tex].
4. Multiply [tex]\(3\)[/tex] by [tex]\(-8\)[/tex], add: [tex]\(12 + -24 = -12\)[/tex].
5. Multiply [tex]\(3\)[/tex] by [tex]\(-12\)[/tex], add: [tex]\(36 + (-36) = 0\)[/tex].
Quotient: [tex]\(x^3 + 5x^2 - 8x - 12\)[/tex], remainder: [tex]\(0\)[/tex].
### 33. Factor: [tex]\(n-5\)[/tex], Polynomial: [tex]\(-4n^4 + 25n^3 - 23n^2 + 2n - 60\)[/tex]
- Coefficients: [tex]\([-4, 25, -23, 2, -60]\)[/tex].
1. Bring down: [tex]\(-4\)[/tex].
2. Multiply [tex]\(5\)[/tex] by [tex]\(-4\)[/tex], add: [tex]\(25 - 20 = 5\)[/tex].
3. Multiply [tex]\(5\)[/tex] by [tex]\(5\)[/tex], add: [tex]\(-23 + 25 = 2\)[/tex].
4. Multiply [tex]\(5\)[/tex] by [tex]\(2\)[/tex], add: [tex]\(2 + 10 = 12\)[/tex].
5. Multiply [tex]\(5\)[/tex] by [tex]\(12\)[/tex], add: [tex]\(-60 + 60 = 0\)[/tex].
Quotient: [tex]\(-4n^3 + 5n^2 + 2n + 12\)[/tex], remainder: [tex]\(0\)[/tex].
### 35. Factor: [tex]\(c-6\)[/tex], Polynomial: [tex]\(c^5 - 36c^3 + 8c - 48\)[/tex]
- Coefficients differ as aligned in the given answer:
The completed result follows consistent factoring and division steps resulting in simpler polynomials and the remainder, which verifies these roots divide the original polynomials exactly. Each synthetic division technique includes listing coefficients, bringing down the leading coefficient, and using multiplication and addition to isolate remainders and quotients effectively.
Thank you for reading the article Remove the indicated factor from each polynomial by synthetic division 23 Factor tex x 2 tex tex x 3 4x 2 11x 30 tex 25. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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