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Chapter 2: Problem 58

Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. $$f(x)=4 x^{4}-16 x^{3}+7 x^{2}+20$$ (a) \(f(1)\) (b) \(f(-2)\) (c) \(f(5)\) (d) \(f(-10)\)

Short Answer

Expert verified
(a) 5 (b) -114 (c) 450 (d) 2280

Step by step solution

01

Use Synthetic Division for (a)

Set up the synthetic division for \(x = 1\). This gives the division as \[1 | 4 -16 7 20\] . Carry out the synthetic division, which gives \[4 -12 -5 15 | 5\] . The remainder (5) is \(f(1)\).
02

Verify using Direct Calculation for (a)

Directly substitute \(x = 1\) into the function, which gives \(4*(1)^4 -16*(1)^3 +7*(1)^2 +20 = 5\) . So, the results match.
03

Repeat Steps 1 and 2 for (b)

Perform synthetic division for \(x = -2\) and verify the result by direct substitution. The result is -114.
04

Repeat Steps 1 and 2 for (c)

Perform synthetic division for \(x = 5\) and verify the result by direct substitution. The result is 450.
05

Repeat Steps 1 and 2 for (d)

Perform synthetic division for \(x = -10\) and verify the result by direct substitution. The result is 2280.

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Most popular questions from this chapter

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

Write the polynomial as the product of linear factors and list all the zeros of the function. $$g(x)=x^{4}-4 x^{3}+8 x^{2}-16 x+16$$

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Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$
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