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Chapter 6: Problem 46

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ r(2 a) $$

Short Answer

Expert verified
\(r(2a) = 24a^3 - 4a^2 - 2.\)

Step by step solution

01

Understand the problem statement

We have been given the polynomial function \( r(x) = 3x^3 - x^2 - 2 \) and need to find the value of \( r(2a) \). This involves substituting \( 2a \) into the polynomial function \( r(x) \) and simplifying the expression.
02

Substitute \( 2a \) into the polynomial

Replace every instance of \( x \) in the function \( r(x) = 3x^3 - x^2 - 2 \) with \( 2a \). This gives us the expression: \[ r(2a) = 3(2a)^3 - (2a)^2 - 2.\]
03

Simplify the expression

Calculate \( (2a)^3 \) which is \( 8a^3 \), and multiply by 3 to get \( 24a^3 \). Next, calculate \( (2a)^2 \) which is \( 4a^2 \). Substitute these back into the expression to get:\[ r(2a) = 24a^3 - 4a^2 - 2.\]
04

Present the simplified expression

The final simplified expression for \( r(2a) \) is: \[ r(2a) = 24a^3 - 4a^2 - 2.\]

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Key Concepts

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

Polynomial Evaluation
Polynomial evaluation is a process where we calculate the value of a polynomial function at a given point or expression.
Polynomials are mathematical expressions involving a sum of powers of variables multiplied by coefficients.
For example, in the polynomial \( r(x) = 3x^3 - x^2 - 2 \), the goal of evaluation is to compute what the value of \( r(x) \) is when \( x \) takes on a specific value.
  • This evaluation can be done by directly substituting the given value or expression into the polynomial.
  • It simplifies calculations, especially when using polynomials to model real-world scenarios and obtain specific data points.
Here, for \( r(2a) \), we replace every instance of \( x \) in \( r(x) \) with \( 2a \), which initiates the evaluation.
Substitution Method
The substitution method in mathematics involves replacing a variable with a given value or expression.
This method is crucial for evaluating polynomials, such as substituting \( x \) with \( 2a \) in the polynomial \( r(x) = 3x^3 - x^2 - 2 \).
  • Identify the variable to substitute, which is \( x \) in this case.
  • Replace each occurrence of \( x \) in the polynomial with the expression \( 2a \).
By doing so, we transform the original polynomial \( r(x) \) into a simpler expression that incorporates the new terms from substitution, making it ready for simplification.
Simplification in Algebra
Simplification in algebra involves reducing expressions into their simplest form.
After substituting \( 2a \) into \( r(x) \), the expression becomes \( r(2a) = 3(2a)^3 - (2a)^2 - 2 \).
  • First, compute powers like \( (2a)^3 \) which results in \( 8a^3 \), and then multiply by 3 to get \( 24a^3 \).
  • Similarly, compute \( (2a)^2 \) giving \( 4a^2 \).
  • Replace these back into the polynomial to obtain \( 24a^3 - 4a^2 - 2 \).
The final expression \( r(2a) = 24a^3 - 4a^2 - 2 \) represents the simplified form, which is now free of unnecessary complexity and ready for interpretation or further use.

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Most popular questions from this chapter

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{3}-4 x^{2} $$

For Exercises \(3-5,\) use the following information. The projected sales of e-books in millions of dollars can be modeled by the function \(S(x)=-17 x^{3}+200 x^{2}-113 x+44,\) where \(x\) is the number of years since 2000 . Use direct substitution to evaluate \(S(8)\)

PERSONAL FINANCE For Exercises \(38-41,\) use the following information. Zach has purchased some home theater equipment for \(\$ 2000,\) which he is financing through the store. He plans to pay \(\$ 340\) per month and wants to have the balance paid off after six months. The formula \(B(x)=2000 x^{6}-\) 340\(\left(x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\) represents his balance after six months if \(x\) represents 1 plus the monthly interest rate (expressed as a decimal). Suppose he finances his purchase at 10.8\(\%\) and plans to pay \(\$ 410\) every month. Will his balance be paid in full after five months?

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{4}+2 x^{3}-8 x-16 ; x+2 $$

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=x^{3}+x^{2}-7 x-3 $$
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