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Chapter 2: Problem 56

Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. $$g(x)=2 x^{6}+3 x^{4}-x^{2}+3$$ (a) \(g(2)\) (b) \(g(1)\) (c) \(g(3)\) (d) \(g(-1)\)

Short Answer

Expert verified
For the given polynomial, \(g(2)\) = 220, \(g(1)\) = 7, \(g(3)\) = 1953, \(g(-1)\) = 24.

Step by step solution

01

Preparing for Synthetic Division

First, arrange the coefficients of the polynomial \(g(x) = 2x^{6}+3x^{4}-x^{2}+3\). Write down the set of coefficients from the highest to the lowest degree terms where the absent terms are represented by 0. This gives us [2, 0, 3, 0, -1, 0, 3]
02

Applying Synthetic Division (g(2))

With \(x = 2\) substitute into the synthetic division process, which gives us [2, 4, 14, 28, 54, 110, 220]. The final number obtained, 220, is the remaining value, which is the solution to \(g(2)\).
03

Applying Synthetic Division (g(1))

This step repeats the process by inserting \(x = 1\) into the Synthetic Division. It gives [2, 2, 5, 5, 4, 4, 7]. The last number, 7, is the value of \(g(1)\).
04

Applying Synthetic Division (g(3))

Repeat the process for \(x = 3\). The results are [2, 6, 24, 72, 216, 650, 1953]. The final number 1953 is \(g(3)\).
05

Applying Synthetic Division (g(-1))

Finally, repeat for \(x = -1\). This generates [2, -2, 6, -6, 12, -12, 24]. The result \(g(-1)\) is 24.
06

Verifying Solution

To verify these results, replace \(x\) by 2, 1, 3, -1 into \(g(x)\) respectively. The results obtained should match with the values obtained from the synthetic division.

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