/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 Evaluate polynomial for \(x=3\) ... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 62

Evaluate polynomial for \(x=3\) and for \(x=-3\). \(2 x^{4}-\frac{1}{9} x^{3}\)

Short Answer

Expert verified
For x = 3, the answer is 159. For x = -3, the answer is 165.

Step by step solution

01

- Substitute x = 3

First, substitute the value of x with 3 in the given polynomial. The polynomial is \(2 x^{4}-\frac{1}{9} x^{3}\).So, it becomes:\[ 2 (3)^{4} - \frac{1}{9} (3)^{3} \]
02

- Evaluate the substituted expression for x = 3

Calculate each term separately:\( (3)^{4} = 81 \)\( 2 (3)^{4} = 2 \times 81 = 162 \)Next, calculate the second term:\( (3)^{3} = 27 \)\( \frac{1}{9} (3)^{3} = \frac{1}{9} \times 27 = 3 \)So, the expression becomes:\( 162 - 3 = 159 \)
03

- Substitute x = -3

Now, substitute the value of x with -3 in the given polynomial:\[ 2 (-3)^{4} - \frac{1}{9} (-3)^{3} \]
04

- Evaluate the substituted expression for x = -3

Calculate each term separately:\( (-3)^{4} = 81 \)\( 2 (-3)^{4} = 2 \times 81 = 162 \)Next, calculate the second term:\( (-3)^{3} = -27 \)\( \frac{1}{9} (-3)^{3} = \frac{1}{9} \times (-27) = -3 \)So, the expression becomes:\( 162 - (-3) = 162 + 3 = 165 \)

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Key Concepts

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

Polynomial Evaluation
Polynomial evaluation refers to the process of finding the value of a polynomial expression for specific values of the variable(s). To evaluate a polynomial, you substitute the given values into the polynomial and simplify. For instance, given the polynomial expression \(2x^4 - \frac{1}{9}x^3\) and the task to evaluate it at \(x=3\) and \(x=-3\), you first replace \(x\) with these values and then perform the arithmetic operations to find the results.
Substitution
Substitution is a core concept in algebra where a variable in an expression is replaced with a specific value. This step converts an algebraic expression into a numerical one, which can then be simplified easily. For the polynomial \(2x^4 - \frac{1}{9}x^3\), substituting \(x=3\) changes it to \(2(3)^4 - \frac{1}{9}(3)^3\). Likewise, substituting \(x=-3\) changes it to \(2(-3)^4 - \frac{1}{9}(-3)^3\). Proper substitution ensures correct and simplified results.
Algebraic Expressions
An algebraic expression consists of variables, numbers, and operations like addition, subtraction, multiplication, and division. In the context of polynomial evaluation, understanding the structure of expressions like \(2x^4 - \frac{1}{9}x^3\) is crucial. In this polynomial, \(2x^4\) and \(-\frac{1}{9}x^3\) are two terms involving a variable raised to different powers and constants multiplied with these terms. Simplification of algebraic expressions involves following the order of operations carefully.
Exponents and Powers
Exponents and powers are fundamental in polynomial expressions. An exponent denotes how many times a number (the base) is multiplied by itself. For example, \((3)^4\) means \(3\) is multiplied by itself three more times (\(3 \times 3 \times 3 \times 3 = 81\)). In evaluating polynomials like \(2x^4 - \frac{1}{9}x^3\), you often deal with powers. First, compute the power (like \((3)^4)\), then multiply by the coefficient (like \(2 \times 81\)), followed by combining all terms. Understanding exponents simplifies the polynomial evaluation process.

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Most popular questions from this chapter

If \(f(x)=c,\) where \(c\) is some positive constant, describe how the graphs of \(y=g(x)\) and \(y=(f+g)(x)\) will differ.

Use the fact that \(10^{3} \approx 2^{10}\) to estimate each of the following powers of \(2 .\) Then compute the power of 2 with a calculator and find the difference between the exact value and the approximation. $$ 2^{14} $$

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