Question

The quotient when \[ 3x^5 - 4x^4 + 5x^3 - 3x^2 + 6x - 8 \] is divided by \( x^2 + x - 3 \) is:

• A. \( 3x^2 - 7x - 21 \)
• B. \( 3x^3 - 7x^2 + 21x - 45 \)
• C. \( 3x^4 - 7x^3 + 21x^2 - 45 + 114 \)
• D. \( 114x - 143 \)

Hint:

To divide polynomials, use long division: divide the leading term, multiply, subtract, and repeat until the degree of the remainder is lower than the divisor.

Answer 1

The Correct Option is B

Step 1: Polynomial Long Division Setup
We divide: \[ 3x^5 - 4x^4 + 5x^3 - 3x^2 + 6x - 8 \] by \[ x^2 + x - 3. \] Step 2: First Division Step
Divide the leading term \( 3x^5 \) by \( x^2 \): \[ \frac{3x^5}{x^2} = 3x^3. \] Multiply: \[ (3x^3) \cdot (x^2 + x - 3) = 3x^5 + 3x^4 - 9x^3. \] Subtract: \[ (3x^5 - 4x^4 + 5x^3) - (3x^5 + 3x^4 - 9x^3) = -7x^4 + 14x^3. \] Step 3: Second Division Step
Divide \( -7x^4 \) by \( x^2 \): \[ \frac{-7x^4}{x^2} = -7x^2. \] Multiply: \[ (-7x^2) \cdot (x^2 + x - 3) = -7x^4 - 7x^3 + 21x^2. \] Subtract: \[ (-7x^4 + 14x^3 - 3x^2) - (-7x^4 - 7x^3 + 21x^2) = 21x^3 - 24x^2. \] Step 4: Third Division Step
Divide \( 21x^3 \) by \( x^2 \): \[ \frac{21x^3}{x^2} = 21x. \] Multiply: \[ % Option (21x) \cdot (x^2 + x - 3) = 21x^3 + 21x^2 - 63x. \] Subtract: \[ (21x^3 - 24x^2 + 6x) - (21x^3 + 21x^2 - 63x) = -45x^2 + 69x. \] Step 5: Fourth Division Step
Divide \( -45x^2 \) by \( x^2 \): \[ \frac{-45x^2}{x^2} = -45. \] Multiply: \[ (-45) \cdot (x^2 + x - 3) = -45x^2 - 45x + 135. \] Subtract: \[ (-45x^2 + 69x - 8) - (-45x^2 - 45x + 135) = 114x - 143. \] Since \( 114x - 143 \) is the remainder, the quotient is: \[ \boxed{3x^3 - 7x^2 + 21x - 45}. \]