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Chapter 5: Problem 10

Evaluate each polynomial for \(x=5\) and \(y=-2.\) $$ x^{2}+5 y^{2}-4 x y $$

Short Answer

Expert verified
The value is 85.

Step by step solution

01

- Substitute the Values

Replace the variables in the polynomial. Substitute \(x=5\) and \(y=-2\) into the polynomial: \[ x^{2}+5 y^{2}-4 x y \] This becomes: \[ 5^{2}+5(-2)^{2}-4(5)(-2) \]
02

- Calculate Each Term

Evaluate each term one by one: First term: \( 5^{2} = 25 \) Second term: \( 5(-2)^{2} = 5(4) = 20 \) Third term: \( -4(5)(-2) = -4(5)(-2) = 40 \)
03

- Combine the Results

Add the results of each term to find the final value: \( 25 + 20 + 40 = 85 \)

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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 crucial technique in algebra to simplify and solve polynomials. It involves replacing the variables in the equation with given numerical values. This allows you to transform a polynomial expression with variables into a numerical expression that is easier to manage.

For example, if we have the polynomial expression: x^{2}+5 y^{2}-4 x y Using the substitution method, we can directly plug in the values of x=5 and y=-2. This transforms the expression to: 5^2+5(-2)^2-4(5)(-2) By doing this, we have transformed a potentially complex algebraic expression into simpler arithmetic operations that are easy to follow and compute.
Polynomial Expressions
Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Understanding the structure of polynomial expressions helps in solving them effectively.

For our example, x^{2}+5 y^{2}-4 x y , we can break it down:
  • First term: x^{2}, where x is the variable
  • Second term: 5 y^{2}, where y is the variable and 5 is the coefficient
  • Third term: -4 x y, where x and y are variables and -4 is the coefficient
Recognizing these components aids in substitution and solving. Poly- means 'many', implying many terms combined together in one expression. Additionally, understanding the standard form and recognizing types of polynomials, like linear, quadratic, and cubic, are helpful.
Steps in Algebra
Algebra involves a systematic approach to manipulate expressions and equations. The steps are usually simple if followed correctly. Here鈥檚 a step-by-step breakdown to solve a polynomial as in our example:

Step 1: Substitute the given values
Replace each variable in the polynomial with its given number. For instance, our polynomial becomes 5^2+5(-2)^2-4(5)(-2)
Step 2: Calculate each term
Evaluate each part step by step:
  • First term: 5^{2} = 25
  • Second term: 5(-2)^{2} = 5(4) = 20
  • Third term: -4(5)(-2) = 40

Step 3: Combine the results
Add all evaluated terms to get the final result: 25 + 20 + 40 = 85

Following these steps methodically ensures accurate results and clarifies the entire process for the learner.

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Most popular questions from this chapter

Find a value of the variable that shows that the two expressions are not equivalent. Answers may vary. $$ 3 x^{2} ;(3 x)^{2} $$

Is it easier to evaluate a polynomial before or after like terms have been combined? Why?

Simplify. $$ y^{4 x} \cdot y^{2 x} $$

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{1}{x-3}\\\ &g(x)=4 x^{3} \end{aligned} $$

Simplify. \(4 a+11-(-2 a-9)\)
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