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Chapter 3: Problem 41

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 \(f\)

Short Answer

Expert verified
The zeros of the function \(f(x)=x^{3}-4x^{2}+x+6\) are \(-1\), \(2\), and \(3\).

Step by step solution

01

Perform Synthetic Division

Apply synthetic division to divide \(x^{3}-4x^{2}+x+6\) by \(x+1\). The number you鈥檒l use for the synthetic division is the zero of \(x+1\), which is \(-1\). Set up \(-1\) and the coefficients of \(f(x)\) to form an array as shown below: \[ \begin{array}{ccc}-1 & \mid & 1 & -4 & 1 & 6 \ & & -1 & 5 & -6 & 6 \\end{array} \]Then proceed to perform operations. Start by bringing down the first coefficient (1 in this case). Multiply \(-1\) by that number and write it beneath the next coefficient. Sum the numbers in this second column to get -1+(-4)=-5. Repeat these steps for the other columns.
02

Identify the Polynomial after Division

After running the synthetic division, you now have the coefficients for the resulting polynomial following the division. The array is as follows: \[ \begin{array}{ccc}-1 & \mid & 1 & -4 & 1 & 6 \ & & -1 & 5 & -6 & 6 \\end{array} \]From the bottom row, the new polynomial is \(x^{2}-5x+6\). Note that you started with a cubic function \(x^{3}-4x^{2}+x+6\) and after dividing by \(x+1\) you got a quadratic function. From the division, there is no remainder.
03

Find All Zeros of f(x)

Now, find the zeros of the function \(f(x)\). From your synthetic division, you have two factors, \(x+1\) and \(x^{2}-5x+6\). Set each factor equal to zero to solve for \(x\). For \(x+1=0\), \(x\) equals \(-1\). For \(x^{2}-5x+6=0\), factor to get \((x-2)(x-3)=0\) and then solve to get \(x=2\), and \(x=3\). Therefore, the zeros of the function \(f(x)\) are \(-1\), \(2\), and \(3\).

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