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Chapter 6: Problem 7

If \(p(x)=2 x^{3}+6 x-12\) and \(q(x)=5 x^{2}+4,\) find each value. 5\([q(2 a)]\)

Short Answer

Expert verified
5[q(2a)] = 100a^2 + 20

Step by step solution

01

Evaluate q(x) at 2a

To find \( q(2a) \), substitute \( x = 2a \) into the function \( q(x) = 5x^2 + 4 \). This gives us:\[ q(2a) = 5(2a)^2 + 4 = 5 \cdot 4a^2 + 4 = 20a^2 + 4 \]
02

Scale the result of q(2a)

Now, we need to find \( 5[q(2a)] \). Use the expression obtained from Step 1:\[ 5[q(2a)] = 5(20a^2 + 4) = 5 \times 20a^2 + 5 \times 4 = 100a^2 + 20 \]

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Key Concepts

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

Understanding Function Evaluation
Function evaluation is a process where we determine the output of a function for a given input. To evaluate a function, we substitute the input value into the variable of the function's formula.
For example, if we have a function \( q(x) = 5x^2 + 4 \), and we want to find \( q(2a) \), we simply place \( 2a \) wherever \( x \) appears in the formula.
This means replacing \( x \) with \( 2a \) and calculating the result: \( q(2a) = 5(2a)^2 + 4 = 20a^2 + 4 \). The logic behind this is that we are checking what the function value is when \( x \) is specifically \( 2a \).
  • Identify the function and the input value/variable.
  • Substitute the input into the function.
  • Simplify the expression to get the result.
Exploring Algebraic Expressions
Algebraic expressions consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They form the backbone of algebra and serve as a way to generalize mathematical operations.
A typical algebraic expression can be seen in our problem as \( 20a^2 + 4 \). Each part of the expression has a special role:
  • Terms: These are the parts separated by addition or subtraction. In \( 20a^2 + 4 \), the terms are \( 20a^2 \) and \( 4 \).
  • Coefficient: This is the numerical part of a term, such as \( 20 \) in \( 20a^2 \).
  • Variable: The letter representing a number, such as \( a \).
  • Exponent: This tells you how many times to multiply the variable by itself, such as \( 2 \) in \( a^2 \).
Understanding how to manipulate these parts allows us to simplify and evaluate expressions effectively.
Diving into Polynomial Arithmetic
Polynomial arithmetic involves operations such as addition, subtraction, multiplication, and division on polynomial expressions. In this exercise, we're primarily focused on multiplication.
When we scaled \( q(2a) \) by multiplying it by 5, we used polynomial multiplication.A polynomial is an expression like \( 5x^2 + 4 \) or \( 2x^3 + 6x - 12 \), which includes variables with non-negative integer exponents and coefficients.Here's how polynomial multiplication works in our example:
  • Your expression is \( 20a^2 + 4 \).
  • Multiply each term by 5:
    • The term \( 20a^2 \) becomes \( 100a^2 \), because \( 5 \times 20a^2 = 100a^2 \).
    • The term \( 4 \) becomes \( 20 \), because \( 5 \times 4 = 20 \).
  • The complete expression is now \( 100a^2 + 20 \).
Understanding these processes will help you tackle various polynomial problems with ease.

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Most popular questions from this chapter

Factor completely. If the polynomial is not factorable, write prime. $$ c^{3}-216 $$

Find the greatest common factor of each set of numbers. $$ 24,84 $$

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 synthetic substitution to estimate the sales for 2008 .

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. $$ f(x)=x^{4}-10 x^{2}+9 $$

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=-x^{3}+6 x^{2}-6 x-5 $$
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