Problem 65 - Elementary and Intermediate Algebra

Question

Evaluate each polynomial for \(a=-2\) and \(b=3 .\) See Example 4. $$ a^{2}+5 a b-b^{2} $$

Step-by-Step Solution

Answer

The evaluated result of the polynomial is \(-35\).

Step 1: Identify the Given Values

We need to evaluate the polynomial \(a^2 + 5ab - b^2\) for the given values \(a = -2\) and \(b = 3\). This means wherever we see \(a\), we substitute it with \(-2\), and wherever we see \(b\), we substitute it with \(3\).

Step 2: Substitute Given Values into the Polynomial

Replace \(a\) with \(-2\) and \(b\) with \(3\) in the polynomial. The expression becomes \((-2)^2 + 5(-2)(3) - (3)^2\).

Step 3: Calculate Each Term Separately

First, calculate each part of the expression separately:- \((-2)^2 = 4\)- \(5(-2)(3) = -30\)- \((3)^2 = 9\)

Step 4: Simplify the Expression

Combine the results from Step 3 to evaluate the polynomial:- Start from the expression: \(4 - 30 - 9\)- First, combine \(4 - 30 = -26\)- Then, combine \(-26 - 9 = -35\)

Step 5: Conclusion

The value of the polynomial \(a^2 + 5ab - b^2\) when \(a = -2\) and \(b = 3\) is \(-35\).

Key Concepts

Substitution Method

When evaluating a polynomial like \(a^2 + 5ab - b^2\), substitution is a key step. The substitution method involves replacing variables in a polynomial with specific values provided in the problem. This technique helps in converting an algebraic expression into a numerical expression that can be easily evaluated.
In this exercise, we are given that \(a = -2\) and \(b = 3\). Thus, we substitute \(a\) with \(-2\) and \(b\) with \(3\) in the polynomial. It's important to ensure that each occurrence of the variables in the polynomial is replaced. When correctly substituted, the polynomial becomes \[(-2)^2 + 5(-2)(3) - (3)^2.\]By focusing on careful substitution, you eliminate the variables, making the expression ready for arithmetic evaluation. The ability to accurately replace variables is essential for correctly evaluating polynomial expressions.

Arithmetic Operations

Once variables are substituted in a polynomial, the next task is evaluating the resulting expression using basic arithmetic operations. This involves performing operations such as addition, subtraction, multiplication, and even exponentiation.
In our example, we have:

• Calculate \((-2)^2\), which equals \(4\). This stems from multiplying \(-2\) by itself.
• For the next term, calculate \(5(-2)(3)\); multiplying these numbers yields \(-30\).
• The last term requires calculating \((3)^2\), resulting in \(9\).

Arithmetic operations require attention to detail, particularly with signs in multiplication and subtraction. Each calculated term should be double-checked to ensure the calculations follow the correct order of operations (PEMDAS/BODMAS). By correctly executing these operations, the polynomial is simplified into a form that can easily be combined.

Polynomial Simplification

After performing arithmetic operations, you arrive at a numeric expression that needs simplification. Polynomial simplification entails combining like terms until the expression cannot be simplified further.
In this instance, we have arrived at the expression:\[4 - 30 - 9.\]To simplify:

• First, calculate \(4 - 30\), which gives \(-26\).
• Next, subtract \(9\) from \(-26\), which results in \(-35\).

The simplified result \(-35\) represents the value of the original polynomial with given values for \(a\) and \(b\). Simplification ensures that your final answer is concise and easy to interpret, providing a clearer understanding of the polynomial's value given the specified conditions.