Problem 32 - Technical Mathematics with Calculus

Question

Challenge Problems. $$\left(x^{a-1}+y^{a-2}\right)\left(x^{a}+y^{a-1}\right)$$

Step-by-Step Solution

Answer

After expanding and simplifying, the expression is \(x^{2a-1} + x^{a-1}y^{a-1} + x^{a}y^{a-2} + y^{2a-3}\).

Step 1: Expand the expression using the distributive property

Distribute each term in the first set of parentheses by each term in the second set of parentheses. This is done by multiplying the first term in the first parentheses by each term in the second parentheses and then doing the same with the second term in the first parentheses.

Step 2: Multiply the terms

Multiply the terms using the laws of exponents. When multiplying terms with the same base, add the exponents. For example, multiplying \(x^{a-1}\) by \(x^a\) results in \(x^{(a-1)+a}=x^{2a-1}\).

Step 3: Combine like terms

After multiplication, see if there are any like terms that can be combined. In this case, there are no like terms as each term contains distinct powers of x and y.

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial when working with algebraic expressions, especially when you're multiplying polynomials. The laws are essentially shortcuts to make calculations easier and faster. For instance, if you have two exponential terms with the same base, you can simply add their exponents when multiplying them together. This is particularly handy in the provided exercise.

Take the multiplication of \(x^{a-1}\) by \(x^a\), for example. According to the laws of exponents, we add the exponents: \((a-1) + a\), which simplifies to \(2a-1\). Why does this work? Because multiplying two powers with the same base is like saying you're multiplying x by itself \(a-1\) times, and then again by itself another \(a\) times — a total of \(2a-1\) times! This simple law is a powerful tool that makes working with exponents much more manageable.

Combining Like Terms

When simplifying algebraic expressions, combining like terms is a key step. Like terms are terms that have exactly the same variable parts raised to the same power. For example, \(2x^2\) and \(5x^2\) are like terms and can be combined to \(7x^2\). However, \(x^2\) and \(x^3\) are not like terms because the exponents are different, reflecting differing powers of x.

In the exercise, you might have noticed that after applying the distributive property and the laws of exponents, no like terms appeared. This often happens when working with polynomials, particularly when each term has different powers. Although it may look neat to try and simplify further, remember that only like terms can be combined. Any attempt to combine terms that are not like would be a misstep in the simplification process.

Multiplication of Polynomials

Multiplying polynomials might seem daunting at first, but with a firm grasp of the distributive property, it becomes much simpler. To multiply polynomials, distribute each term of the first polynomial over each term of the second polynomial, and vice versa. This means you'll be using the distributive property several times in one problem.

Let's look at the example from the exercise. You have two polynomials, \(x^{a-1}+y^{a-2}\) and \(x^a+y^{a-1}\). The key is to multiply every term by every other term: \(x^{a-1}\) gets multiplied by \(x^a\) and \(y^{a-1}\), then \(y^{a-2}\) gets multiplied by \(x^a\) and \(y^{a-1}\). This will result in four terms, which you can then inspect for any like terms to combine. Remember, it's critical you apply the laws of exponents accurately during this step to ensure that the result is correct.