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Determine the quotient and remainder for \((4x^3-11x-9) \div (2x-3), x \neq \frac{3}{2}\)

Notice the dividend has no \(x^2\) term. When terms are missing in the dividend or divisor, placeholders are used to keep the terms properly aligned.

Determine the quotient and remainder for \((4x^3-11x-9) \div (2x-3), x \neq \frac{3}{2}\)

The quotient is \(2x^2+3x-1\) and the remainder is \(-12\). Therefore,

\[\frac{4x^3-11x-9}{2x-3}=2x^2+3x-1-\frac{12}{2x-3}, x \neq \frac{3}{2}\]

Determine \(\dfrac{3x^4+12x^2-2x^3-5}{x^2-1}, x \neq \pm 1\).

When setting up the division, ensure the terms of the dividend and divisor are in descending order, and use placeholders for missing terms.

Determine \(\dfrac{3x^4+12x^2-2x^3-5}{x^2-1}, x \neq \pm 1\).

The quotient is \(3x^2-2x+15\) and the remainder is \(2x-10\). Therefore,

\[\frac{3x^4-2x^3+12x^2-5}{x^2-1}=3x^2-2x+15+\frac{-2x+10}{x^2-1}, x \neq \pm 1\]

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