Ask a New Question
Menu

Using long division, determine which of the following polynomials divides evenly for a divisor of x+4 .(1 point) Responses 3x2−19x−28 3 x squared minus 19 x minus 28 - no response given −3x2−19x−28 negative 3 x squared minus 19 x minus 28 - no response given −3x2−19x+28 negative 3 x squared minus 19 x plus 28 - no response given 3x2+19x−28

1 answer

To determine which of the given polynomials divides evenly by \( x + 4 \) using polynomial long division, we can take each polynomial and divide it by \( x + 4 \).

Step 1: Dividing \( 3x^2 - 19x - 28 \) by \( x + 4 \)

  1. Divide the leading term of the dividend by the leading term of the divisor: \[ \frac{3x^2}{x} = 3x \]
  2. Multiply \( 3x \) by \( x + 4 \): \[ 3x(x + 4) = 3x^2 + 12x \]
  3. Subtract from the original polynomial: \[ (3x^2 - 19x - 28) - (3x^2 + 12x) = -31x - 28 \]
  4. Now divide \(-31x\) by \(x\): \[ \frac{-31x}{x} = -31 \]
  5. Multiply \(-31\) by \(x + 4\): \[ -31(x + 4) = -31x - 124 \]
  6. Subtract: \[ (-31x - 28) - (-31x - 124) = 96 \]

The remainder is \(96\), so \(3x^2 - 19x - 28\) does not divide evenly by \(x + 4\).

Step 2: Dividing \(-3x^2 - 19x - 28\) by \(x + 4\)

  1. Divide the leading term: \[ \frac{-3x^2}{x} = -3x \]
  2. Multiply: \[ -3x(x + 4) = -3x^2 - 12x \]
  3. Subtract: \[ (-3x^2 - 19x - 28) - (-3x^2 - 12x) = -7x - 28 \]
  4. Now divide \(-7x\): \[ \frac{-7x}{x} = -7 \]
  5. Multiply: \[ -7(x + 4) = -7x - 28 \]
  6. Subtract: \[ (-7x - 28) - (-7x - 28) = 0 \]

The remainder is \(0\). Therefore, \(-3x^2 - 19x - 28\) divides evenly by \(x + 4\).

Step 3: Dividing \(-3x^2 - 19x + 28\) by \(x + 4\)

  1. Leading term: \[ \frac{-3x^2}{x} = -3x \]
  2. Multiply: \[ -3x(x + 4) = -3x^2 - 12x \]
  3. Subtract: \[ (-3x^2 - 19x + 28) - (-3x^2 - 12x) = -7x + 28 \]
  4. Now divide \(-7x\): \[ \frac{-7x}{x} = -7 \]
  5. Multiply: \[ -7(x + 4) = -7x - 28 \]
  6. Subtract: \[ (-7x + 28) - (-7x - 28) = 56 \]

The remainder is \(56\), so \(-3x^2 - 19x + 28\) does not divide evenly by \(x + 4\).

Step 4: Dividing \(3x^2 + 19x - 28\) by \(x + 4\)

  1. Leading term: \[ \frac{3x^2}{x} = 3x \]
  2. Multiply: \[ 3x(x + 4) = 3x^2 + 12x \]
  3. Subtract: \[ (3x^2 + 19x - 28) - (3x^2 + 12x) = 7x - 28 \]
  4. Now divide \(7x\): \[ \frac{7x}{x} = 7 \]
  5. Multiply: \[ 7(x + 4) = 7x + 28 \]
  6. Subtract: \[ (7x - 28) - (7x + 28) = -56 \]

The remainder is \(-56\), so \(3x^2 + 19x - 28\) does not divide evenly by \(x + 4\).

Conclusion: The only polynomial among the given options that divides evenly by \(x + 4\) is:

\[ \boxed{-3x^2 - 19x - 28} \]