Problem 41 - College Algebra

Question

Use synthetic division to divide $$ f(x)=x^{3}-4 x^{2}+x+6 \text { by } x+1 $$ Use the result to find all zeros of \(f\)

Step-by-Step Solution

Answer

The zeros of \(f(x)\) are -1, 2 and 3.

Step 1: Setup Synthetic Division

Set up the synthetic division with -1 (the root corresponding to the divisor \(x + 1\)) on the left and the coefficients of the cubic polynomial on the right. Write -1 in the left side and coefficients 1 (coefficient for \(x^3\)), -4 (coefficient for \(x^2\)), 1 (coefficient for \(x\)), and 6 (constant term) on the right side of the vertical line.

Step 2: Execute Synthetic Division

Perform synthetic division by bringing down the first coefficient (1 for \(x^3\)), multiplying -1 by this coefficient and placing the product under the next coefficient (-4), adding to get new coefficient, and repeating the process until you reach the end of the row. This should give the coefficients of the resulting quotient polynomial.

Step 3: Write Down The Quotient Polynomial

After doing the synthetic division, the resulting polynomial's coefficients are on the bottom row. Starting with the degree of the polynomial that is one less than the original polynomial, write out the quotient polynomial, \(q(x)\). Exclusive of any remainder carried from the division, the final resulting polynomial after synthetic division is: \(q(x) = x^2 - 5x + 6\).

Step 4: Find The Zeros Of The Quotient Polynomial

Solve \(q(x) = x^2 - 5x + 6\) equals to zero for its roots, which are also zeros of the original polynomial, \(f(x)\). Add the root of the divisor that we got at the beginning of synthetic division, which is -1, to the found roots.