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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
The function values are: (a) \(g(2)=251\), (b) \(g(1)=10\), (c) \(g(3)=5745\), and (d) \(g(-1)=5\). The results have been verified by replacing the x-values directly into the function.

Step by step solution

01

Evaluate g(x) at x=2 using synthetic division

Write the coefficients of the polynomial and arrange them in an array: [2, 0, 3, 0, -1, 0, 3]. The zeros fill in for the missing terms of the polynomial \(g(x)\) (i.e., \(x^5\), \(x^3\), and \(x\)). Then, begin synthetic division with \(x=2\). The result is: [2, 4, 8, 16, 31, 62, 124, 251]. The last digit, 251, is the value of \(g(2)\).
02

Evaluate g(x) at x=1 using synthetic division

Perform synthetic division with \(x=1\). The result is: [2, 2, 5, 5, 4, 4, 7, 10]. The last digit, 10, is the value of \(g(1)\).
03

Evaluate g(x) at x=3 using synthetic division

Perform synthetic division with \(x=3\). The result is: [2, 6, 24, 72, 213, 639, 1914, 5745]. The last digit, 5745, is the value of \(g(3)\).
04

Evaluate g(x) at x=-1 using synthetic division

Perform synthetic division with \(x=-1\). The result is: [2, -2, 2, -2, 4, -4, 8, 5]. The last digit, 5, is the value of \(g(-1)\).
05

Verification

To verify the results, evaluate values directly into the polynomial. Recognize that \(g(x)\) reduces to addition of constant numbers when \(x\) is replaced with respective values. Verification confirms the obtained results.

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