Question

question 1 (2 points)
complete the synthetic division table and find the coefficient listed below.
$(x^4 - 5x^3 - 8x^2 + 13x - 12) \div (x - 6)$

the coefficient of $x^2$ is
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Explanation:

Step1: Identify the root and coefficients

For synthetic division of \((x^{4}-5x^{3}-8x^{2}+13x - 12)\div(x - 6)\), the root is \(r = 6\) (since \(x-6=0\) gives \(x = 6\)). The coefficients of the dividend polynomial are \(1\) (for \(x^{4}\)), \(-5\) (for \(x^{3}\)), \(-8\) (for \(x^{2}\)), \(13\) (for \(x\)), and \(-12\) (constant term).

Step2: Set up the synthetic division table

The first row (coefficients of dividend) is \(1\), \(-5\), \(-8\), \(13\), \(-12\). The root \(6\) is placed outside.

Step3: Bring down the first coefficient

Bring down the first coefficient \(1\) to the third row (result row). So the first number in the third row is \(1\).

Step4: Multiply and add for \(x^{3}\) term

Multiply the brought - down number (\(1\)) by the root (\(6\)): \(1\times6 = 6\). Add this to the second coefficient (\(-5\)): \(-5+6=1\). This is the coefficient for the \(x^{3}\) term in the quotient.

Step5: Multiply and add for \(x^{2}\) term

Multiply the result from the previous step (\(1\)) by the root (\(6\)): \(1\times6 = 6\). Add this to the third coefficient (\(-8\)): \(-8 + 6=-2\). This is the coefficient of \(x^{2}\) in the quotient.

Answer:

The coefficient of \(x^{2}\) is \(-2\)