Problem 9 - Elementary Algebra

Question

Factor completely. $$ 16 x 2-9 $$

Step-by-Step Solution

Answer

The expression factors to \((4x - 3)(4x + 3)\).

Step 1: Identify the Expression Type

The expression is a difference of squares, which follows the pattern \(a^2 - b^2\). Here, \(16x^2\) can be written as \((4x)^2\) and \(9\) as \(3^2\).

Step 2: Apply Difference of Squares Formula

Using the formula \(a^2 - b^2 = (a - b)(a + b)\), identify \(a = 4x\) and \(b = 3\). Substitute these back into the formula to get \((4x - 3)(4x + 3)\).

Step 3: Check the Factorization

Verify the factorization by expanding \((4x - 3)(4x + 3)\) to ensure it equals the original expression. This results in \((4x)^2 - (3)^2 = 16x^2 - 9\), confirming the factorization is correct.

Key Concepts

Difference of Squares

The concept of the "difference of squares" is an important algebraic identity that frequently appears in equations and expressions. It is based on the pattern that for any two values, say \(a\) and \(b\), the expression \(a^2 - b^2\) can be factored as \((a - b)(a + b)\). This is known as the difference of squares identity. The usefulness of this identity arises because it helps us break down complex quadratic expressions into two linear factors, making them easier to handle. To visualize why this works, let's consider the expansion:

• You distribute \((a - b)\) and \((a + b)\) to get \(a^2 + ab - ab - b^2\).
• Notice that the middle terms \(+ab\) and \(-ab\) cancel each other out.
• What remains is \(a^2 - b^2\), proving our factorization pattern.

This means any time you see a term structured as a square subtracting another square, you can break it down neatly using this rule.
Understanding and spotting the difference of squares is vital for factoring algebraic expressions effectively.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients arranged in terms of powers. They take the general form: \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n\), \(a_{n-1}\), ..., \(a_0\) are coefficients and \(x\) represents the variable. Each term is separated by either a plus or a minus sign.In mathematics, polynomial expressions are everywhere. They describe everything from basic algebraic problems to complex physical phenomena. The standard algebraic polynomial can be manipulated through methods like addition, subtraction, multiplication, and division, but also through factoring, which simplifies them for problem-solving.Factoring polynomials, such as the exercise exemplifies, often involves recognizing patterns like the difference of squares. This initial insight allows one to factorize expressions into simpler parts, just like identifying \(16x^2 - 9\) as \((4x)^2 - 3^2\). It then becomes easier to manage or solve the polynomial for given values of \(x\).
Understanding polynomial expressions and their characteristics is the key to unlocking many algebraic puzzles.

Expanding Expressions

Expanding expressions refers to the process of multiplying out the terms of an expression to eliminate any parenthesis, thereby writing it as a sum or difference of terms. For example, expanding \((4x - 3)(4x + 3)\) means calculating the product of these expressions to see if it matches its unfactored form.To expand, you take each term in the first parenthesis and multiply it by each term in the second, using the distributive property. For the given expression:

• Multiply \(4x\) by \(4x\) to get \(16x^2\).
• Multiply \(4x\) by \(+3\) to get \(+12x\).
• Multiply \(-3\) by \(4x\) to get \(-12x\).
• Multiply \(-3\) by \(+3\) to get \(-9\).

The middle terms cancel out \(+12x\) and \(-12x\), leaving \(16x^2 - 9\), which confirms the expression has been correctly expanded back to its original form.
This process is essential in algebra as it demonstrates that one can both "factor" and "expand" expressions reliably. Mastering these skills allows for a deeper understanding of manipulating equations in varied mathematical contexts.