Problem 84 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expanded form is \(9x^{2n} - 42x^n + 49\).

Step 1: Recognize the Binomial to Expand

The given expression is \((3x^n - 7)^2\). This is a binomial raised to a power of 2, suggesting we need to use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).

Step 2: Identify the Components

In the expression \((3x^n - 7)^2\), identify the components as \(a = 3x^n\) and \(b = 7\).

Step 3: Apply the Binomial Formula

Using the binomial square formula: \[ (3x^n - 7)^2 = (3x^n)^2 - 2(3x^n)(7) + 7^2 \]

Step 4: Calculate the Square of Each Term

First, calculate \((3x^n)^2\): \((3x^n)^2 = 9x^{2n}\). Second, calculate \(7^2\): \(7^2 = 49\).

Step 5: Calculate the Cross Product Term

Calculate \(2 \times (3x^n) \times 7\): \(2 \times (3x^n) \times 7 = 42x^n\).

Step 6: Combine All the Terms

Combine all the calculated parts to get the complete expanded form: \(9x^{2n} - 42x^n + 49\).

Key Concepts

Exponents

Exponents are a powerful mathematical concept that helps simplify repeated multiplication. When you see a number or variable raised to a power, like in \( x^n \), the exponent \( n \) tells you how many times to multiply the base \( x \) by itself.
For example, \( x^3 \) means \( x \times x \times x \).

• The base is the number or expression that is multiplied.
• The exponent indicates the number of times the base is used as a factor.
• An exponent of 1 means the base remains unchanged, and an exponent of 0 means the result is always 1, due to the rules of exponents.

In the context of the expression \( 3x^n \), \( x^n \) suggests that \( x \) is multiplied by itself \( n \) times. Multiplying \( 3x^n \) by itself during expansion leverages these properties to compute \( (3x^n)^2 = 9x^{2n} \). This understanding is crucial for expanding binomials correctly using exponents.

Binomial Theorem

The binomial theorem is a central concept in algebra for expanding expressions of the form \((a+b)^n\). It allows us to expand powers of binomials without direct multiplication, which is especially helpful for large exponents. However, this exercise specifically dealt with using the binomial square formula as a shortcut for squares.
When faced with \((a-b)^2\), rather than calculating \((a-b)(a-b)\), you apply the formula:

• \(a^2\)
• \(-2ab\)
• \(+b^2\)

These components make the expression easier to handle and remember, notably important for variables in polynomials (like \(3x^n\)) or constants (like 7).
Using the binomial theorem in this way helps us systematically expand any binomial expression, harnessing the power of algebraic identities.

Polynomials

Polynomials are expressions made up of terms that each include a variable raised to a non-negative integer power, multiplied by a coefficient. For instance,

• A term like \(3x^n\) is part of a polynomial because it includes an exponent (\(n\)) and a coefficient (3).
• The expression obtained from the original exercise, \(9x^{2n} - 42x^n + 49\), is itself a polynomial.

Each term of the polynomial contributes to its degree, which is the highest exponent present in the expression. In our expanded form, the degree is \(2n\) because that's the highest exponent on \(x\).
A clear understanding of polynomials, including how to add, subtract, and multiply them, is essential for manipulating these algebraic structures effectively. Being comfortable with polynomials makes it easier to understand and work with more complex algebraic expressions in mathematics.