Problem 81 - Algebra for College Students

Question

Find the indicated products. Assume all variables that appear as exponents represent positive integers. $$\left(x^{2 a}-7\right)\left(x^{2 a}-3\right)$$

Step-by-Step Solution

Answer

The result is \( x^{4a} - 10x^{2a} + 21 \).

Step 1: Understand the expression

We need to calculate the product of two binomials: \((x^{2a} - 7)\) and \((x^{2a} - 3)\). This means we will use the distributive property to expand this expression.

Step 2: Apply the distributive property

Using the distributive property, we expand the expression:\[ (x^{2a} - 7)(x^{2a} - 3) = x^{2a}(x^{2a} - 3) - 7(x^{2a} - 3) \].Apply distribution to each term in the binomials.

Step 3: Expand the first term

Distribute the first term \(x^{2a}\) over the second binomial:\[ x^{2a} \cdot x^{2a} - x^{2a} \cdot 3 = x^{4a} - 3x^{2a} \].

Step 4: Expand the second term

Distribute the second term \(-7\) over the second binomial:\[ -7 \cdot x^{2a} + (-7)\cdot (-3) = -7x^{2a} + 21 \].

Step 5: Combine like terms

Now combine the results of your distributions:\[ x^{4a} - 3x^{2a} - 7x^{2a} + 21\].Combine like terms \( -3x^{2a} \) and \( -7x^{2a} \):\[ x^{4a} - 10x^{2a} + 21 \].

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to multiply a single term by terms inside a parenthesis. It is a handy tool for expanding expressions. When you apply the distributive property, you ensure that each term inside the first set of parentheses is multiplied by each term inside the second set.
In the original exercise, we used it to expand the product \((x^{2a} - 7)(x^{2a} - 3)\). Here's how it works step-by-step:

• Multiply the first term from the first binomial by each term in the second binomial.
• Then, multiply the second term from the first binomial by each term in the second binomial.
• As a result, you end up with separate terms that you can then simplify further.

This property is crucial when dealing with polynomials, as it simplifies expressions and helps to solve equations easily.

Binomial Expansion

Binomial expansion refers to expanding expressions that consist of two terms. In algebra, a binomial is simply a polynomial with two terms. To expand a binomial, you can use the distributive property.
In the exercise, the binomials were \((x^{2a} - 7)\) and \((x^{2a} - 3)\). By expanding them, you explore all possible combinations of the multiplication of these two binomials:

• The first term with the first term: \(x^{2a} \cdot x^{2a} = x^{4a}\)
• The first term with the second term: \(x^{2a} \cdot (-3) = -3x^{2a}\)
• The second term with the first term: \(-7 \cdot x^{2a} = -7x^{2a}\)
• The second term with the second term: \(-7 \cdot (-3) = 21\)

Binomial expansion allows a structured way to ensure no term is overlooked during distribution, facilitating further simplification.

Exponents

Exponents are used in mathematics to denote repeated multiplication of a base number. The notation \(a^n\) signifies that \(a\), the base, is multiplied by itself \(n\) times. Understanding how to manipulate exponents is vital when dealing with terms like \(x^{2a}\) or \(x^{4a}\) in our exercise.
The rules of exponents make multiplication and simplification of expressions possible:

• Multiplication of like bases: \(x^m \cdot x^n = x^{m+n}\)
• Power of a power: \((x^m)^n = x^{m \times n}\)

In the step-by-step process, for example, \(x^{2a} \cdot x^{2a} = x^{4a}\), due to the fact we added the exponents since the base \(x\) was the same.

Polynomial Multiplication

Polynomial multiplication extends the concepts of binomial expansion and the distributive property into a broader range of expressions. Polynomials are expressions that can have multiple terms, possibly with varying powers and coefficients, and multiplying them involves a systematic approach to ensure each term is correctly distributed.
In the given exercise, we focused on multiplying two binomials, a specific type of polynomial multiplication:

• Identify each term: Acknowledge each term involved in the polynomials.
• Distribute accurately: For example, multiply each term in the first polynomial by all terms in the second polynomial.
• Combine like terms: Finally, simplify the expression by adding coefficients for terms with the same power, as seen when we combined \(-3x^{2a}\) and \(-7x^{2a}\) to form \(-10x^{2a}\).

This systematic approach ensures accuracy and completeness, with each step leading to a simpler form of the original polynomial expression.