Asked by Nanika Zoldyck
Use long division to divide the polynomial 15x2+14x−8 by 3x+4. What is the quotient?(1 point)
Responses
5x−2+163x+4
5 x minus 2 plus Start Fraction 16 over 3 x plus 4 End Fraction
5x+2−163x+4
5 x plus 2 minus Start Fraction 16 over 3 x plus 4 End Fraction
5x−2
5 x minus 2
5x+2
Responses
5x−2+163x+4
5 x minus 2 plus Start Fraction 16 over 3 x plus 4 End Fraction
5x+2−163x+4
5 x plus 2 minus Start Fraction 16 over 3 x plus 4 End Fraction
5x−2
5 x minus 2
5x+2
Answers
Answered by
GPT-4o mini
To divide the polynomial \(15x^2 + 14x - 8\) by \(3x + 4\) using long division, follow these steps:
1. **Divide the leading term of the dividend by the leading term of the divisor**:
\[
\frac{15x^2}{3x} = 5x
\]
So, the first term of the quotient is \(5x\).
2. **Multiply the entire divisor by \(5x\)** and subtract from the original polynomial:
\[
(3x + 4)(5x) = 15x^2 + 20x
\]
Now, subtract this from the original polynomial:
\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]
3. **Repeat the process with the new polynomial \(-6x - 8\)**. Again, divide the leading term:
\[
\frac{-6x}{3x} = -2
\]
Now add \(-2\) to the quotient.
4. **Multiply the divisor by \(-2\)**:
\[
(3x + 4)(-2) = -6x - 8
\]
Now subtract this from \(-6x - 8\):
\[
(-6x - 8) - (-6x - 8) = 0
\]
5. **Putting it all together**: The quotient is \(5x - 2\) and there is no remainder.
Thus, the final quotient when dividing \(15x^2 + 14x - 8\) by \(3x + 4\) is:
\[
\boxed{5x - 2}
\]
1. **Divide the leading term of the dividend by the leading term of the divisor**:
\[
\frac{15x^2}{3x} = 5x
\]
So, the first term of the quotient is \(5x\).
2. **Multiply the entire divisor by \(5x\)** and subtract from the original polynomial:
\[
(3x + 4)(5x) = 15x^2 + 20x
\]
Now, subtract this from the original polynomial:
\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = 14x - 20x - 8 = -6x - 8
\]
3. **Repeat the process with the new polynomial \(-6x - 8\)**. Again, divide the leading term:
\[
\frac{-6x}{3x} = -2
\]
Now add \(-2\) to the quotient.
4. **Multiply the divisor by \(-2\)**:
\[
(3x + 4)(-2) = -6x - 8
\]
Now subtract this from \(-6x - 8\):
\[
(-6x - 8) - (-6x - 8) = 0
\]
5. **Putting it all together**: The quotient is \(5x - 2\) and there is no remainder.
Thus, the final quotient when dividing \(15x^2 + 14x - 8\) by \(3x + 4\) is:
\[
\boxed{5x - 2}
\]
Answered by
Nanika Zoldyck
Which of the following expressions is a factor of 2x2+9x−35 ? Use long division to solve.(1 point)
Responses
x+5
x plus 5
2x−5
2 x minus 5
x−7
x minus 7
2x+7
Responses
x+5
x plus 5
2x−5
2 x minus 5
x−7
x minus 7
2x+7
Answered by
GPT-4o mini
To determine which expression is a factor of the polynomial \(2x^2 + 9x - 35\), we will perform polynomial long division for each possible factor candidate against the polynomial.
### Step 1: Testing \(x + 5\)
1. **Setup for long division**: Divide \(2x^2 + 9x - 35\) by \(x + 5\).
2. **First term**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{2x^2}{x} = 2x
\]
3. **Multiply and subtract**:
- Multiply \(x + 5\) by \(2x\):
\[
(x + 5)(2x) = 2x^2 + 10x
\]
- Subtract:
\[
(2x^2 + 9x - 35) - (2x^2 + 10x) = 9x - 10x - 35 = -x - 35
\]
4. **Repeat the process**:
- Divide \(-x\) by \(x\):
\[
\frac{-x}{x} = -1
\]
- Multiply and subtract:
\[
(x + 5)(-1) = -x - 5
\]
- Subtract:
\[
(-x - 35) - (-x - 5) = -35 + 5 = -30
\]
Since we are left with \(-30\) (not zero), \(x + 5\) is **not a factor**.
### Step 2: Testing \(2x - 5\)
1. **First term**:
\[
\frac{2x^2}{2x} = x
\]
2. **Multiply and subtract**:
- Multiply \(2x - 5\) by \(x\):
\[
(2x - 5)(x) = 2x^2 - 5x
\]
- Subtract:
\[
(2x^2 + 9x - 35) - (2x^2 - 5x) = 9x + 5x - 35 = 14x - 35
\]
3. **Repeat**:
- Divide \(14x\) by \(2x\):
\[
\frac{14x}{2x} = 7
\]
- Multiply and subtract:
\[
(2x - 5)(7) = 14x - 35
\]
- Subtract:
\[
(14x - 35) - (14x - 35) = 0
\]
Since the remainder is \(0\), \(2x - 5\) **is a factor**.
### Conclusion
The factor of \(2x^2 + 9x - 35\) is:
\[
\boxed{2x - 5}
\]
### Step 1: Testing \(x + 5\)
1. **Setup for long division**: Divide \(2x^2 + 9x - 35\) by \(x + 5\).
2. **First term**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{2x^2}{x} = 2x
\]
3. **Multiply and subtract**:
- Multiply \(x + 5\) by \(2x\):
\[
(x + 5)(2x) = 2x^2 + 10x
\]
- Subtract:
\[
(2x^2 + 9x - 35) - (2x^2 + 10x) = 9x - 10x - 35 = -x - 35
\]
4. **Repeat the process**:
- Divide \(-x\) by \(x\):
\[
\frac{-x}{x} = -1
\]
- Multiply and subtract:
\[
(x + 5)(-1) = -x - 5
\]
- Subtract:
\[
(-x - 35) - (-x - 5) = -35 + 5 = -30
\]
Since we are left with \(-30\) (not zero), \(x + 5\) is **not a factor**.
### Step 2: Testing \(2x - 5\)
1. **First term**:
\[
\frac{2x^2}{2x} = x
\]
2. **Multiply and subtract**:
- Multiply \(2x - 5\) by \(x\):
\[
(2x - 5)(x) = 2x^2 - 5x
\]
- Subtract:
\[
(2x^2 + 9x - 35) - (2x^2 - 5x) = 9x + 5x - 35 = 14x - 35
\]
3. **Repeat**:
- Divide \(14x\) by \(2x\):
\[
\frac{14x}{2x} = 7
\]
- Multiply and subtract:
\[
(2x - 5)(7) = 14x - 35
\]
- Subtract:
\[
(14x - 35) - (14x - 35) = 0
\]
Since the remainder is \(0\), \(2x - 5\) **is a factor**.
### Conclusion
The factor of \(2x^2 + 9x - 35\) is:
\[
\boxed{2x - 5}
\]
Answered by
Nanika Zoldyck
Which of the following expressions is not a factor of 3x3−10x2−143x−90 ? Use long division to solve.(1 point)
Responses
x−9
x minus 9
x−6
x minus 6
3x+2
3 x plus 2
x+5
Responses
x−9
x minus 9
x−6
x minus 6
3x+2
3 x plus 2
x+5
Answered by
GPT-4o mini
To determine which expression is **not** a factor of the polynomial \(3x^3 - 10x^2 - 143x - 90\), we will perform polynomial long division for each candidate factor against the polynomial.
### Step 1: Testing \(x - 9\)
1. **Set up for long division**: Divide \(3x^3 - 10x^2 - 143x - 90\) by \(x - 9\).
2. **First term**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{3x^3}{x} = 3x^2
\]
3. **Multiply and subtract**:
- Multiply \(x - 9\) by \(3x^2\):
\[
(x - 9)(3x^2) = 3x^3 - 27x^2
\]
- Subtract:
\[
(3x^3 - 10x^2 - 143x - 90) - (3x^3 - 27x^2) = (-10x^2 + 27x^2) - 143x - 90 = 17x^2 - 143x - 90
\]
4. **Next term**: Divide \(17x^2\) by \(x\):
\[
\frac{17x^2}{x} = 17x
\]
5. **Multiply and subtract**:
- Multiply \(x - 9\) by \(17x\):
\[
(x - 9)(17x) = 17x^2 - 153x
\]
- Subtract:
\[
(17x^2 - 143x - 90) - (17x^2 - 153x) = (-143x + 153x) - 90 = 10x - 90
\]
6. **Next term**: Divide \(10x\) by \(x\):
\[
\frac{10x}{x} = 10
\]
7. **Multiply and subtract**:
- Multiply \(x - 9\) by \(10\):
\[
(x - 9)(10) = 10x - 90
\]
- Subtract:
\[
(10x - 90) - (10x - 90) = 0
\]
Since the remainder is \(0\), \(x - 9\) **is a factor**.
---
### Step 2: Testing \(x - 6\)
1. **Set up for long division**: Divide \(3x^3 - 10x^2 - 143x - 90\) by \(x - 6\).
2. **First term**:
\[
\frac{3x^3}{x} = 3x^2
\]
3. **Multiply and subtract**:
- Multiply \(x - 6\) by \(3x^2\):
\[
(x - 6)(3x^2) = 3x^3 - 18x^2
\]
- Subtract:
\[
(3x^3 - 10x^2 - 143x - 90) - (3x^3 - 18x^2) = (-10x^2 + 18x^2) - 143x - 90 = 8x^2 - 143x - 90
\]
4. **Next term**:
\[
\frac{8x^2}{x} = 8x
\]
5. **Multiply and subtract**:
- Multiply \(x - 6\) by \(8x\):
\[
(x - 6)(8x) = 8x^2 - 48x
\]
- Subtract:
\[
(8x^2 - 143x - 90) - (8x^2 - 48x) = (-143x + 48x) - 90 = -95x - 90
\]
6. **Next term**:
\[
\frac{-95x}{x} = -95
\]
7. **Multiply and subtract**:
- Multiply \(x - 6\) by \(-95\):
\[
(x - 6)(-95) = -95x + 570
\]
- Subtract:
\[
(-95x - 90) - (-95x + 570) = -90 - 570 = -660
\]
Since the remainder is \(-660\) (not zero), \(x - 6\) **is not a factor**.
---
### Conclusion
The expression that is **not** a factor of \(3x^3 - 10x^2 - 143x - 90\) is:
\[
\boxed{x - 6}
\]
### Step 1: Testing \(x - 9\)
1. **Set up for long division**: Divide \(3x^3 - 10x^2 - 143x - 90\) by \(x - 9\).
2. **First term**: Divide the leading term of the dividend by the leading term of the divisor:
\[
\frac{3x^3}{x} = 3x^2
\]
3. **Multiply and subtract**:
- Multiply \(x - 9\) by \(3x^2\):
\[
(x - 9)(3x^2) = 3x^3 - 27x^2
\]
- Subtract:
\[
(3x^3 - 10x^2 - 143x - 90) - (3x^3 - 27x^2) = (-10x^2 + 27x^2) - 143x - 90 = 17x^2 - 143x - 90
\]
4. **Next term**: Divide \(17x^2\) by \(x\):
\[
\frac{17x^2}{x} = 17x
\]
5. **Multiply and subtract**:
- Multiply \(x - 9\) by \(17x\):
\[
(x - 9)(17x) = 17x^2 - 153x
\]
- Subtract:
\[
(17x^2 - 143x - 90) - (17x^2 - 153x) = (-143x + 153x) - 90 = 10x - 90
\]
6. **Next term**: Divide \(10x\) by \(x\):
\[
\frac{10x}{x} = 10
\]
7. **Multiply and subtract**:
- Multiply \(x - 9\) by \(10\):
\[
(x - 9)(10) = 10x - 90
\]
- Subtract:
\[
(10x - 90) - (10x - 90) = 0
\]
Since the remainder is \(0\), \(x - 9\) **is a factor**.
---
### Step 2: Testing \(x - 6\)
1. **Set up for long division**: Divide \(3x^3 - 10x^2 - 143x - 90\) by \(x - 6\).
2. **First term**:
\[
\frac{3x^3}{x} = 3x^2
\]
3. **Multiply and subtract**:
- Multiply \(x - 6\) by \(3x^2\):
\[
(x - 6)(3x^2) = 3x^3 - 18x^2
\]
- Subtract:
\[
(3x^3 - 10x^2 - 143x - 90) - (3x^3 - 18x^2) = (-10x^2 + 18x^2) - 143x - 90 = 8x^2 - 143x - 90
\]
4. **Next term**:
\[
\frac{8x^2}{x} = 8x
\]
5. **Multiply and subtract**:
- Multiply \(x - 6\) by \(8x\):
\[
(x - 6)(8x) = 8x^2 - 48x
\]
- Subtract:
\[
(8x^2 - 143x - 90) - (8x^2 - 48x) = (-143x + 48x) - 90 = -95x - 90
\]
6. **Next term**:
\[
\frac{-95x}{x} = -95
\]
7. **Multiply and subtract**:
- Multiply \(x - 6\) by \(-95\):
\[
(x - 6)(-95) = -95x + 570
\]
- Subtract:
\[
(-95x - 90) - (-95x + 570) = -90 - 570 = -660
\]
Since the remainder is \(-660\) (not zero), \(x - 6\) **is not a factor**.
---
### Conclusion
The expression that is **not** a factor of \(3x^3 - 10x^2 - 143x - 90\) is:
\[
\boxed{x - 6}
\]