Questions: Divide. Check your answer by showing that the product of the divisor and the quotient, plus the remainder, is the dividend. (64 x^3 - 27) / (4 x - 3)

Solution Steps

To solve the given division problem, we will use polynomial long division. We will divide the polynomial \(64x^3 - 27\) by \(4x - 3\). After obtaining the quotient and remainder, we will verify our result by multiplying the divisor by the quotient and adding the remainder to check if we get back the original dividend.

We need to divide the polynomial \(64x^3 - 27\) by \(4x - 3\). Using polynomial long division, we find the quotient and remainder.

The quotient of the division is \(16x^2 + 12x + 9\) and the remainder is \(0\).

To verify the result, we multiply the divisor \(4x - 3\) by the quotient \(16x^2 + 12x + 9\) and add the remainder \(0\): \[ (4x - 3)(16x^2 + 12x + 9) + 0 \]

Simplifying the expression: \[ (4x - 3)(16x^2 + 12x + 9) = 4x \cdot 16x^2 + 4x \cdot 12x + 4x \cdot 9 - 3 \cdot 16x^2 - 3 \cdot 12x - 3 \cdot 9 \] \[ = 64x^3 + 48x^2 + 36x - 48x^2 - 36x - 27 \] \[ = 64x^3 - 27 \]

Final Answer