Problem 28 - Beginning and Intermediate Algebra

Question

Evaluate each polynomial when \(x=-4\) and \(y=3\) $$y^{2}-x^{2}$$

Step-by-Step Solution

Answer

When \(x=-4\) and \(y=3\), the polynomial \(y^2 - x^2\) evaluates to \[-7\].

Step 1: Substitute given values into the polynomial

First, substitute \(x=-4\) and \(y=3\) into the polynomial \(y^2 - x^2\). It will look like this: \[(3)^2 - (-4)^2\]

Step 2: Square the values

Now, square the values of x and y which means multiplying them by themselves. It will result in: \[9 - 16\]

Step 3: Simplify the expression

Finally, simplify the expression by subtracting 16 from 9. \[9 - 16 = -7\] The result of the polynomial when \(x=-4\) and \(y=3\) is -7.

Key Concepts

Substitution

Substitution is the first step in evaluating polynomials. It involves replacing variables with specific numbers. In our example, the polynomial is \(y^2 - x^2\). For evaluation purposes, we substitute \(x = -4\) and \(y = 3\). This effectively transforms the polynomial into a numerical expression: \((3)^2 - (-4)^2\). To substitute correctly:

• Identify the variables in the polynomial.
• Replace each variable with the given value.

Substitution allows us to convert a variable-based expression into numerical form, paving the way for further calculations.

Squaring Numbers

Squaring a number means multiplying it by itself. In algebra, this operation is crucial for simplifying expressions with exponents.
In our example, we need to square \(3\) and \(-4\). This results in \((3)^2 = 9\) and \((-4)^2 = 16\). Notice how squaring \(-4\) results in a positive 16. This is because:

• Negative times negative yields a positive product.
• Any number, positive or negative, squared will produce a non-negative result.

Understanding how to square numbers correctly is essential for performing calculations on polynomials.

Simplification

Simplification is the process of making an expression easier to understand by reducing it to a more basic form. Following substitution and squaring in our polynomial, we obtain a numerical expression \(9 - 16\).

• First, identify like terms which can be combined.
• Perform appropriate arithmetic operations, such as addition or subtraction.

Here, we subtract 16 from 9, which simplifies to \(-7\). Simplification is important as it provides the final, most comprehensible version of our expression.

Negative Numbers

Working with negative numbers, especially in polynomials, requires careful attention to sign. In our exercise, we had to evaluate \((3)^2 - (-4)^2\). Here:

• The squares of numbers, whether negative or positive, result in non-negative outcomes.
• However, subtracting with a negative number leads to changes in sign that must be noted: subtracting a larger positive from a smaller one yields a negative result.

Recognizing the intricacies of negative numbers in subtraction ensures accuracy in evaluation, leading to the correct solution of \(-7\) for this exercise.