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Chapter 5: Problem 57

Evaluate polynomial for \(x=3\) and for \(x=-3\). \(2 x^{2}-3 x+7\)

Short Answer

Expert verified
The value is 16 for x = 3 and 34 for x = -3.

Step by step solution

01

Understand the Polynomial

The given polynomial is 2x^2 - 3x + 7.
02

Substitute x = 3

First, substitute x = 3 into the polynomial. Calculate 2(3)^2 - 3(3) + 7.
03

Calculate Each Term for x = 3

Calculate the value of each term: 2(3)^2 = 18, -3(3) = -9, and add 7.
04

Sum the Results for x = 3

Now find the sum: 18 - 9 + 7. The result is 16.
05

Substitute x = -3

Next, substitute x = -3 into the polynomial. Calculate 2(-3)^2 - 3(-3) + 7.
06

Calculate Each Term for x = -3

Calculate the value of each term: 2(-3)^2 = 18, -3(-3) = 9, and add 7.
07

Sum the Results for x = -3

Finally, find the sum: 18 + 9 + 7. The result is 34.

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Key Concepts

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

substitution method
The substitution method is a way to evaluate polynomials by replacing the variable with a given number. In our exercise, this method involves substituting the values 3 and -3 into the polynomial. This means wherever you see the variable x in the polynomial, you replace it with the given number. For example, replacing x with 3 in the polynomial 2x^2 - 3x + 7 results in 2(3)^2 - 3(3) + 7. This process helps you convert the expression into a numerical calculation, simplifying the evaluation.
polynomials
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. In our case, the polynomial is 2x^2 - 3x + 7. Polynomials are made up of terms, which are separated by plus (+) or minus (-) signs. Each term includes a coefficient (like 2 or -3), a variable (like x), and an exponent (like 2 in x^2). Polynomials can represent a wide variety of mathematical relationships and are used frequently in algebra and calculus. Understanding polynomials is key to solving many types of math problems.
simplification
Simplification involves reducing a mathematical expression to its simplest form. For polynomials, this means combining like terms and performing arithmetic operations to get a single, simplified expression. In our example, after substituting x with 3 or -3, we simplified each term individually. For x = 3, calculate each term: 2(3)^2 becomes 18, -3(3) becomes -9, and adding 7 remains 7. Finally, sum the results (18 - 9 + 7). For x = -3, perform the same steps. Simplification makes it easier to evaluate and understand the polynomial.
exponents
Exponents are a way to represent repeated multiplication of the same number. In the polynomial 2x^2 - 3x + 7, the exponent 2 in x^2 indicates that x is multiplied by itself: x*x. Evaluating exponents is an important step in polynomial evaluation. For example, with x = 3, x^2 becomes 3*3, which equals 9. Then, multiply by the coefficient: 2(3^2) becomes 2*9, which is 18. Understanding how to work with exponents helps you accurately evaluate each term of the polynomial.

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Most popular questions from this chapter

Simplify. Write \(8^{-3} \cdot 32 \div 16^{2}\) as a power of 2

For each pair of functions \(f\) and \(g,\) determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=\frac{2 x}{x+1}\\\ &g(x)=2 x+5 \end{aligned} $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=5+x\\\ &g(x)=6-2 x \end{aligned} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=3 x^{2}\\\ &g(x)=\frac{1}{x-9} \end{aligned} $$

To prepare for Section \(5.2,\) review operations with integers (Sections \(1.5-1.7)\) Perform the indicated operations. $$ -3(5) $$
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