Problem 26 - College Algebra

Question

Divide using synthetic division. $$\left(x^{2}-6 x-6 x^{3}+x^{4}\right) \div(6+x)$$

Step-by-Step Solution

Answer

The result of the synthetic division is \(x^3 - 12x^2 + 72x + 73\).

Step 1: Set Up the Problem

Rewrite the divisor \(6 + x\) as \(x + 6\) because it's typically written with the higher power variable first. Arrange the coefficients of dividend polynomial \(x^{4}-6 x^{3}-6 x^2+x\) which are \[1, -6, -6, 1\]. Then, write down the zero of the divisor by setting \(x + 6 = 0\), which gives \(x = -6\).

Step 2: Perform Synthetic Division

Organize your findings in a synthetic division setup. You should have -6 on the outside and \[1, -6, -6, 1\] on the inside. Bring down the leading coefficient (in this case, 1) to start your solution row. For each remaining number in the dividend row, multiply by the zero from step 1 and add to the next number in the original row. In this case, it would be: \[(-6)\times1=-6, (-6)-6=-12, (-6)\times(-12)=72, 72+1=73.\] Your solution row should be \[1, -12, 72, 73\].

Step 3: Write the Result

Interpret your solution row as coefficients of a polynomial. Because you started with a polynomial of degree 4 and divided by a polynomial of degree 1, you should end with a polynomial of degree 3. The result from your solution row should be: \(x^3 - 12x^2 + 72x + 73\).