Asked by sayletyler
Divide the polynomial 12x2+2x−9 by 4x−2. Then express the polynomial as the product of the divisor and the quotient plus the remainder. (1 point) Responses 12x2+2x−9=(4x−2)(3x+2)−13 12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren minus 13 12x2+2x−9=(4x−2)(3x+2)+5 12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren plus 5 12x2+2x−9=(4x−2)(3x+2)−5 12 x squared plus 2 x minus 9 is equal to open paren 4 x minus 2 close paren times open paren 3 x plus 2 close paren minus 5 12x2+2x−9=(4x−2)(3x−1)−11
Answers
Answered by
sayletyler
The image shows the first few steps of the polynomial division (3x4+9x2−13)÷(x2−5x). Complete the division process and find the remainder of this polynomial division. (1 point) Responses The remainder is 407x. The remainder is 407 x . The remainder is 420x−13. The remainder is 420 x minus 13 . The remainder is 407. The remainder is 407. The remainder is 2,087.
Answered by
GPT-4o mini
To find the quotient and remainder when dividing the polynomial \(12x^2 + 2x - 9\) by \(4x - 2\), we can use polynomial long division.
### Steps to Divide:
1. **Divide the leading term of the dividend by the leading term of the divisor**:
\[
\frac{12x^2}{4x} = 3x
\]
2. **Multiply the entire divisor by this result**:
\[
3x(4x - 2) = 12x^2 - 6x
\]
3. **Subtract this from the original polynomial**:
\[
(12x^2 + 2x - 9) - (12x^2 - 6x) = 2x + 6x - 9 = 8x - 9
\]
4. **Now repeat the process with the new polynomial \(8x - 9\)**:
\[
\frac{8x}{4x} = 2
\]
5. **Multiply the divisor by this result**:
\[
2(4x - 2) = 8x - 4
\]
6. **Subtract this from \(8x - 9\)**:
\[
(8x - 9) - (8x - 4) = -9 + 4 = -5
\]
### Final Results:
The quotient is \(3x + 2\) and the remainder is \(-5\).
Thus, we can express the polynomial as:
\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]
### Confirming the Response:
The correct expression based on the calculations is:
\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]
Therefore, the correct answer is:
**12x² + 2x - 9 = (4x - 2)(3x + 2) - 5**.
### Steps to Divide:
1. **Divide the leading term of the dividend by the leading term of the divisor**:
\[
\frac{12x^2}{4x} = 3x
\]
2. **Multiply the entire divisor by this result**:
\[
3x(4x - 2) = 12x^2 - 6x
\]
3. **Subtract this from the original polynomial**:
\[
(12x^2 + 2x - 9) - (12x^2 - 6x) = 2x + 6x - 9 = 8x - 9
\]
4. **Now repeat the process with the new polynomial \(8x - 9\)**:
\[
\frac{8x}{4x} = 2
\]
5. **Multiply the divisor by this result**:
\[
2(4x - 2) = 8x - 4
\]
6. **Subtract this from \(8x - 9\)**:
\[
(8x - 9) - (8x - 4) = -9 + 4 = -5
\]
### Final Results:
The quotient is \(3x + 2\) and the remainder is \(-5\).
Thus, we can express the polynomial as:
\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]
### Confirming the Response:
The correct expression based on the calculations is:
\[
12x^2 + 2x - 9 = (4x - 2)(3x + 2) - 5
\]
Therefore, the correct answer is:
**12x² + 2x - 9 = (4x - 2)(3x + 2) - 5**.
Answered by
GPT-4o mini
To complete the division of the polynomial \(3x^4 + 9x^2 - 13\) by \(x^2 - 5x\), we will perform polynomial long division step by step.
### Step-by-Step Polynomial Division:
1. **Divide the leading term**:
\[
\frac{3x^4}{x^2} = 3x^2
\]
2. **Multiply the entire divisor by \(3x^2\)**:
\[
3x^2(x^2 - 5x) = 3x^4 - 15x^3
\]
3. **Subtract from the original polynomial**:
\[
(3x^4 + 9x^2 - 13) - (3x^4 - 15x^3) = 15x^3 + 9x^2 - 13
\]
4. **Repeat with the new polynomial, \(15x^3 + 9x^2 - 13\)**:
- Divide the leading term:
\[
\frac{15x^3}{x^2} = 15x
\]
- Multiply the divisor:
\[
15x(x^2 - 5x) = 15x^3 - 75x^2
\]
- Subtract:
\[
(15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 9x^2 + 75x^2 - 13 = 84x^2 - 13
\]
5. **Repeat with the new polynomial, \(84x^2 - 13\)**:
- Divide the leading term:
\[
\frac{84x^2}{x^2} = 84
\]
- Multiply the divisor:
\[
84(x^2 - 5x) = 84x^2 - 420x
\]
- Subtract:
\[
(84x^2 - 13) - (84x^2 - 420x) = 420x - 13
\]
So at this point, we have divided completely, and the next divisor \(420x - 13\) has a degree smaller than \(x^2 - 5x\).
### Conclusion:
Thus, the remainder of the polynomial division is:
\[
\text{The remainder is } 420x - 13.
\]
Therefore, the correct response is: **The remainder is \(420x - 13\)**.
### Step-by-Step Polynomial Division:
1. **Divide the leading term**:
\[
\frac{3x^4}{x^2} = 3x^2
\]
2. **Multiply the entire divisor by \(3x^2\)**:
\[
3x^2(x^2 - 5x) = 3x^4 - 15x^3
\]
3. **Subtract from the original polynomial**:
\[
(3x^4 + 9x^2 - 13) - (3x^4 - 15x^3) = 15x^3 + 9x^2 - 13
\]
4. **Repeat with the new polynomial, \(15x^3 + 9x^2 - 13\)**:
- Divide the leading term:
\[
\frac{15x^3}{x^2} = 15x
\]
- Multiply the divisor:
\[
15x(x^2 - 5x) = 15x^3 - 75x^2
\]
- Subtract:
\[
(15x^3 + 9x^2 - 13) - (15x^3 - 75x^2) = 9x^2 + 75x^2 - 13 = 84x^2 - 13
\]
5. **Repeat with the new polynomial, \(84x^2 - 13\)**:
- Divide the leading term:
\[
\frac{84x^2}{x^2} = 84
\]
- Multiply the divisor:
\[
84(x^2 - 5x) = 84x^2 - 420x
\]
- Subtract:
\[
(84x^2 - 13) - (84x^2 - 420x) = 420x - 13
\]
So at this point, we have divided completely, and the next divisor \(420x - 13\) has a degree smaller than \(x^2 - 5x\).
### Conclusion:
Thus, the remainder of the polynomial division is:
\[
\text{The remainder is } 420x - 13.
\]
Therefore, the correct response is: **The remainder is \(420x - 13\)**.
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