To divide the polynomials \( (8x^4 - 12x^3 + 12x^2 - 4x + 7) \) by \( (4x^2 - 2x) \), we can use polynomial long division.
-
Setup the division: Write the division in long division format. \[ \begin{array}{r|l} 4x^2 - 2x & 8x^4 - 12x^3 + 12x^2 - 4x + 7 \end{array} \]
-
Divide the leading term: Divide the leading term \( 8x^4 \) by \( 4x^2 \) to get \( 2x^2 \).
-
Multiply and subtract: Multiply \( 2x^2 \) by \( (4x^2 - 2x) \) and subtract it from the original polynomial. \[ 2x^2 \cdot (4x^2 - 2x) = 8x^4 - 4x^3 \] Subtract: \[ (8x^4 - 12x^3) - (8x^4 - 4x^3) = -8x^3 \]
-
Bring down the next term: Bring down the next term from the original polynomial: \[ -8x^3 + 12x^2 \]
-
Repeat the division: Divide \( -8x^3 \) by \( 4x^2 \) to get \( -2x \).
-
Multiply and subtract: Multiply \( -2x \) by \( (4x^2 - 2x) \): \[ -2x \cdot (4x^2 - 2x) = -8x^3 + 4x^2 \] Subtract: \[ (-8x^3 + 12x^2) - (-8x^3 + 4x^2) = 8x^2 \]
-
Bring down the next term: Bring down the next term: \[ 8x^2 - 4x \]
-
Repeat the division: Divide \( 8x^2 \) by \( 4x^2 \) to get \( 2 \).
-
Multiply and subtract: Multiply \( 2 \) by \( (4x^2 - 2x) \): \[ 2 \cdot (4x^2 - 2x) = 8x^2 - 4x \] Subtract: \[ (8x^2 - 4x) - (8x^2 - 4x) = 0 \]
-
Result: Since the remainder is 0, the division completes successfully. The quotient is: \[ 2x^2 - 2x + 2 \]
Thus, the result of dividing \( (8x^4 - 12x^3 + 12x^2 - 4x + 7) \) by \( (4x^2 - 2x) \) is: \[ 2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x} \]
So the full answer is: \[ 2x^2 - 2x + 2 + \frac{7}{4x^2 - 2x} \]