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Chapter 5: Problem 9

Evaluate each polynomial for \(x=5\) and \(y=-2.\) $$ x^{2}-3 y^{2}+2 x y $$

Short Answer

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Step by step solution

01

- Substitute the values

First, substitute the given values of variables into the polynomial. So, replace every instance of \(x\) with \(5\) and \(y\) with \(-2\). The expression becomes: \[ 5^{2} - 3(-2)^{2} + 2(5)(-2) \]
02

- Calculate each term

Next, calculate the value of each term separately: 1. \( 5^{2} = 25 \) 2. \(-2\right)^{2} = (-2) \times (-2) = 4 \) 3. \( -3 \times 4 = -12 \)4. \( 2 \times 5 \times -2 = -20 \)
03

- Combine the terms

Now, add and subtract all the calculated values together: \[ 25 - 12 - 20 \]
04

- Simplify the final expression

Finally, simplify the expression by combining the terms: \[ 25 - 12 = 13 \]\[ 13 - 20 = -7 \]

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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 variables with given values.
For example, in the polynomial:
When evaluating a polynomial, you replace every instance of the variable with the specified value.
In our exercise, we were given the polynomial
To evaluate this using the substitution method, we replaced each instance of 'x' with 5 and 'y' with -2:
The substitution method can be broken down into a few steps:
  • Identify the given values for the variables.
  • Replace the variables in the polynomial with the given values.
  • Calculate each term separately.
  • Combine the calculated values to get the final result.
algebraic expressions
Algebraic expressions are combinations of numbers, variables, and operations.
They can represent a variety of mathematical relationships.
Each term in an algebraic expression is made up of a constant, a variable, or a combination of both along with mathematical operations like addition, subtraction, multiplication, and division.
For instance:
In the given polynomial Here, each term 鈥� such as Understanding algebraic expressions involves interpreting and manipulating these terms to solve problems.
This is crucial when evaluating polynomials or simplifying expressions.
simplifying polynomials
Simplifying polynomials involves combining like terms to reduce the expression to its most basic form.
Like terms are terms that have the same variables raised to the same power.
Here鈥檚 the basic approach to simplifying polynomials:
  • Identify and group like terms in the polynomial.
  • Combine the coefficients of the like terms.
  • Simplify any numerical calculations.
In our example:
We started by evaluating each term individually:
Finally, we combined all the calculated terms to get our simplified result:
By breaking down the polynomial into simpler parts and then combining them, we make complex expressions easier to handle and understand.

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Most popular questions from this chapter

To prepare for Section 5.3, review combining like terms and evaluating expressions (Sections 1.6 and 1.8). Combine like terms. $$ 5 a-7 b-8 a+b $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 25^{8} \square 125^{5} $$

Simplify. $$ \left[\left(5^{-3}\right)^{2}\right]^{-1} $$

Review factoring expressions and solving equations. $$ 3 x-1=0 $$

Using the window \([-5,5,-1,9],\) graph \(y_{1}=5\) \(y_{2}=x+2,\) and \(y_{3}=\sqrt{x} .\) Then predict what shape the graphs of \(y_{1}+y_{2}, y_{1}+y_{3},\) and \(y_{2}+y_{3}\) will take. Use a graph to check each prediction.
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