Question

find the quotient and remainder using either long division or synthetic division.
\frac{x^{4}-5 x^{3}+21 x - 4}{x - 3}
the quotient is
the remainder is
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Explanation:

Step1: Set up synthetic division

Write the coefficients of the polynomial \(x^{4}-5x^{3}+0x^{2}+21x - 4\) (the coefficient of \(x^{2}\) is \(0\)) and the zero of the divisor \(x - 3\) (which is \(3\)).
The coefficients are \(1,-5,0,21,-4\).

Step2: Perform synthetic - division

• Bring down the first coefficient \(1\).
• Multiply \(1\) by \(3\) to get \(3\), add to \(-5\) gives \(-2\).
• Multiply \(-2\) by \(3\) to get \(-6\), add to \(0\) gives \(-6\).
• Multiply \(-6\) by \(3\) to get \(-18\), add to \(21\) gives \(3\).
• Multiply \(3\) by \(3\) to get \(9\), add to \(-4\) gives \(5\).

The coefficients of the quotient polynomial (degree is one less than the dividend) are \(1,-2,-6,3\). So the quotient polynomial is \(x^{3}-2x^{2}-6x + 3\) and the remainder is \(5\).

Answer:

The quotient is \(x^{3}-2x^{2}-6x + 3\), the remainder is \(5\).