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Chapter 7: Q37. (page 351)

If $p (x) = 3 x^{2} - 2 x + 5 and r (x) = x^{3} + x + 1$ then find $r (x + 1) - r (x^{2})$ .

Short Answer

Expert verified

The value of $r (x + 1) - r (x^{2}) = - x^{6} + x^{3} + 2 x^{2} + 4 x + 2$ .

Step by step solution

01

Step 1. Write down the given information.

The given functions are $p (x) = 3 x^{2} - 2 x + 5 and r (x) = x^{3} + x + 1$ .

02

Step 2. Calculation.

Evaluating $r (x + 1) - r (x^{2})$ by plugging $x = x + 1 and x = x^{2}$ in given function $r (x) = x^{3} + x + 1$ and then subtracting $r (x^{2})$ from $r (x + 1)$ .

Therefore,

$\begin{matrix} r (x) = x^{3} + x + 1 .... (Given) \\ r (x + 1) = {(x + 1)}^{3} + (x + 1) + 1 \\ = x^{3} + 1 + 3 x^{2} + 3 x + x + 1 + 1 \\ = x^{3} + 3 x^{2} + 4 x + 3.... (1) \\ Now, r (x^{2}) = {(x^{2})}^{3} + (x^{2}) + 1 \\ = x^{6} + x^{2} + 1.... (2) \end{matrix}$

Subtracting (2) from (1) gives,

$\begin{matrix} r (x + 1) 鈭� r (x^{2}) = (x^{3} + 3 x^{2} + 4 x + 3) 鈭� (x^{6} + x^{2} + 1) \\ = 鈭� x^{6} + x^{3} + 2 x^{2} + 4 x + 2 \end{matrix}$

03

Step 3. Conclusion.

The value of $r (x + 1) - r (x^{2}) = - x^{6} + x^{3} + 2 x^{2} + 4 x + 2$ .

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Most popular questions from this chapter

Complete each of the following.

Graph each function by making a table of values.
Determine consecutive values of x between which each real zero is located.
Estimate the x-coordinate at which the relative maxima and relative minima occur.

$f (x) = x^{3} - 4 x^{2} + 2 x - 1$

Complete each of the following.

Graph each function by making a table of values.
Determine consecutive values of x between which each real zero is located.
Estimate the x-coordinate at which the relative maxima and relative minima occur.

f (x) = x^{3} - 2 x^{2} + 6

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b. Determine whether it represents an odd-degree or an even-degree polynomial function, and

c. state the number of real zeros.

EMPLOYMENT For Exercises 27鈥�30, use the graph that models the unemployment rates from 1975鈥�2000.

Describe the turning points and end behaviour of the graph.

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