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Chapter 3: Problem 45

Given \(f(x)=2 x^{4}-5 x^{3}+x^{2}-7\) a. Evaluate \(f(4)\) b. Determine the remainder when \(f(x)\) is divided by \((x-4)\)

Short Answer

Expert verified
a) 201b) 201

Step by step solution

01

Substitute 4 into the function

To evaluate the function at a specific point, substitute the value of 4 into the function. This means calculating:e(4) = 2(4)^4 - 5(4)^3 + (4)^2 - 7
02

Simplify the exponents

First, calculate the exponents:ewline(4)^4 = 256 ewline(4)^3 = 64 ewline(4)^2 = 16
03

Perform the multiplications

Next, multiply these values by their respective coefficients:2(256) = 512 - 5(64) = 320 + 1(16) = 16 - 7
04

Evaluate the expression

Now, evaluate the expression:512 - 320 + 16 - 7 = 201
05

Remainder theorem application

According to the Remainder Theorem, the remainder when a polynomial function f(x) is divided by (x-a) is simply f(a). Therefore, f(4) gives the remainder.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Remainder Theorem
The Remainder Theorem is a fundamental principle in algebra that relates polynomial division to function evaluation. If you divide a polynomial function, \(f(x)\), by a linear divisor like \((x-a)\), the remainder of this division is simply \(f(a)\). So, if you want to find the remainder without performing the polynomial division, you just need to evaluate the function at \(a\). This can make many problems much simpler and quicker to solve.
Evaluating Functions
Evaluating a function means calculating its value for a specific input. In the function \(f(x) = 2x^{4} - 5x^{3} + x^{2} - 7\), to evaluate \(f(4)\), you simply plug in 4 wherever you see an \(x\). Follow these steps:
  • First, replace each \(x\) with 4: \(2(4)^{4} - 5(4)^{3} + (4)^{2} - 7\).
  • Then solve the exponents: \((4)^{4} = 256\), \((4)^{3} = 64\), \((4)^{2} = 16\).
  • Multiply these values by their coefficients: \(2 \cdot 256 = 512\), \(-5 \cdot 64 = -320\), \(1 \cdot 16 = 16\).
  • Add and subtract these results: \(512 - 320 + 16 - 7\).
  • The final result is \(201\).
So, \(f(4) = 201\).
Substitution Method
The substitution method is a straightforward way to evaluate a polynomial. You substitute each variable with a specific value and then simplify step by step.
For example, in the polynomial \(f(x) = 2x^{4} - 5x^{3} + x^{2} - 7\), if we want to find \(f(4)\), we substitute and simplify:
  • First, substitute 4 for \(x\): \(2(4)^{4} - 5(4)^{3} + (4)^{2} - 7\).
  • Simplify the powers of 4: \((4)^4 = 256\), \((4)^3 = 64\), \((4)^2 = 16\).
  • Continue evaluating by performing any multiplications: \(2 \cdot 256 = 512\), \(-5 \cdot 64 = -320\), \(+1 \cdot 16 = 16\).
  • Combine the results: \(512 - 320 + 16 - 7 = 201\).
Once you carry out each step, you can quickly determine the value of the polynomial at a specific point.
Polynomial Division
Polynomial division is a method of dividing one polynomial by another, similar to long division with numbers. However, when working with the Remainder Theorem, you can often avoid the full division process.
Here鈥檚 why:
The Remainder Theorem tells us that to find the remainder of a polynomial \(f(x)\) divided by \((x-a)\), you just need to evaluate \(f(a)\).
For example, to find the remainder of \(2x^4 - 5x^3 + x^2 - 7\) by \((x-4)\), just evaluate \(f(4)\):
  • Calculate \(f(4) = 2(4)^4 - 5(4)^3 + (4)^2 - 7\).
  • Expanding and simplifying gives us the remainder, which is 201.
Using the Remainder Theorem simplifies many polynomial tasks and can save a lot of time compared to performing a full polynomial division.

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Most popular questions from this chapter

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{3 x-5}-4<0 $$

Given \(y=f(x)\) a. Divide the numerator by the denominator to write \(f(x)\) in the form \(f(x)=\) quotient \(+\frac{\text { remainder }}{\text { divisor }}\). b. Use transformations of \(y=\frac{1}{x}\) to graph the function. $$ f(x)=\frac{5 x+11}{x+2} $$

Determine if the statement is true or false. a. Use the intermediate value theorem to show that \(f(x)=\) \(2 x^{2}-7 x+4\) has a real zero on the interval [2,3] . b. Find the zeros.

A monthly phone plan costs \(\$ 59.95\) for unlimited texts and 400 min. If more than 400 min are used the plan charges an addition \(\$ 0.45\) per minute. For \(x\) minutes used, the average cost per minute \(\overline{C_{1}}(x)\) (in \$) is given by $$\overline{C_{1}}(x)=\left\\{\begin{array}{ll} \frac{59.95}{x} & \text { for } 0 \leq x \leq 400 \\ \frac{59.95+0.45(x-400)}{x} & \text { for } x>400 \end{array}\right.$$ a. Find the average cost if a customer talks for \(252 \mathrm{~min}\) \(366 \mathrm{~min},\) and \(400 \mathrm{~min} .\) Round to 2 decimal places. b. Find the average cost if a customer talks for \(436 \mathrm{~min}\) \(582 \mathrm{~min},\) and \(700 \mathrm{~min}\). Round to 2 decimal places. c. Suppose a second phone plan costs \(\$ 79.95\) per month for unlimited minutes and unlimited texts. Write an average cost function to represent the average cost \(\overline{C_{2}}(x)\) (in \$) for \(x\) minutes used. d. Find the average cost \(\overline{C_{2}}(x)\) for the second plan for \(252 \mathrm{~min}, 400 \mathrm{~min},\) and \(700 \mathrm{~min}\)

a. Determine if the upper bound theorem identifies the given number as an upper bound for the real zeros of \(f(x)\). b. Determine if the lower bound theorem identifies the given number as a lower bound for the real zeros of \(f(x)\). \(f(x)=3 x^{5}-16 x^{4}+5 x^{3}+90 x^{2}-138 x+36\) a. 6 b. -3
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