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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

5 answers

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: \[ \boxed{5x - 2} \]

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

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: \[ \boxed{2x - 5} \]

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

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: \[ \boxed{x - 6} \]