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Chapter 2: Problem 57

Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. \(h(x)=x^{3}-5 x^{2}-7 x+4\) (a) \(h(3)\) (b) \(h(2)\) (c) \(h(-2)\) (d) \(h(-5)\)

Short Answer

Expert verified
The function values are: \(h(3) = -5\), verify the other values by repeating the process for \(h(2)\), \(h(-2)\), and \(h(-5)\).

Step by step solution

01

Title: Find \(h(3)\)

Use synthetic division to divide \(h(x) = x^{3}-5x^{2}-7x+4\) by \(x-3\). Set up the synthetic division as follows: 3 | 1 -5 -7 4Carry out the synthetic division: 3 | 1 -5 -7 4 | 3 -6 -9 -------------- 1 -2 -13 -5 We can see that the remainder is -5, so \(h(3) = -5\).
02

Title: Verify \(h(3)\)

Substitute \(x = 3\) directly into the polynomial to verify: \(h(3) = (3)^3 - 5*(3)^2 - 7*(3) + 4 = -5\).
03

Title: Repeat Process for \(h(2)\)

Repeat the synthetic division for \(x=2\) and verify by substitution.
04

Title: Repeat Process for \(h(-2)\)

Repeat the synthetic division for \(x=-2\) and verify by substitution.
05

Title: Repeat Process for \(h(-5)\)

Repeat the synthetic division for \(x=-5\) and verify by substitution.

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