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Chapter 2: Problem 55

Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. \(f(x)=2 x^{3}-7 x+3\) (a) \(f(1)\) (b) \(f(-2)\) (c) \(f\left(\frac{1}{2}\right)\) (d) \(f(2)\)

Short Answer

Expert verified
After using synthetic division, \(f(1) = -2\), \(f(-2) = 13\), \(f(\frac{1}{2}) = 2.75\), and \(f(2) = 9\). These results can be verified by substituting these values back into the function.

Step by step solution

01

Apply synthetic division with divisor 1

Perform synthetic division of the function \(f(x) = 2x^{3} - 7x + 3\) using the value \(x = 1\). The remainder you get from the synthetic division is \(f(1)\).
02

Apply synthetic division with divisor -2

Repeat the process in step 1 but now using \(x= -2\). The remainder you get from the synthetic division is \(f(-2)\).
03

Apply synthetic division with divisor \(\frac{1}{2}\)

Now, for \(x = \frac{1}{2}\), perform synthetic division with it as a factor. The remainder from this synthetic division would give you \(f(\frac{1}{2})\).
04

Apply synthetic division with divisor 2

Lastly, for \(x = 2\), perform synthetic division again. You get \(f(2)\) as the remainder of this operation.
05

Verification

Verify the obtained results by directly substituting the \(x\) values into the polynomial function and check if they match the corresponding synthetic division remainders.

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