Problem 32 - College Algebra

Question

For the following exercises, multiply the binomials. $$(9 a-4)(9 a+4)$$

Step-by-Step Solution

Answer

\(81a^2 - 16\)

Step 1: Recognize the Identity

The expression given is in the form \((a-b)(a+b)\), which resembles the difference of squares identity: \((a-b)(a+b) = a^2 - b^2\). In this problem, we have \(a = 9a\) and \(b = 4\).

Step 2: Apply the Identity

Now apply the difference of squares identity:\[(9a - 4)(9a + 4) = (9a)^2 - 4^2\]

Step 3: Calculate Each Term

Calculate \((9a)^2\) and \(4^2\):\[(9a)^2 = 81a^2\]\[4^2 = 16\]

Step 4: Subtract the Squares

Subtract the square of \(4\) from the square of \(9a\):\[81a^2 - 16\]

Step 5: State the Final Result

Thus, the product of multiplying the binomials \((9a-4)(9a+4)\) is:\[81a^2 - 16\]

Key Concepts

Difference of Squares

The difference of squares is a special algebraic identity. It's widely used in simplifying expressions and solving equations. The identity is given by

• \((a-b)(a+b) = a^2 - b^2\)

To recognize if you can use the difference of squares, look for two terms in the form \((a - b)(a + b)\). Here, the identities form a product of two conjugate terms.
In the given exercise,

• \(a = 9a\) and
• \(b = 4\)

operate within this special identity's pattern. This gives the resulting expression

• \((9a)^2 - 4^2\)

Understanding this helps quickly simplify and solve expressions without lengthy multiplication steps. Thus,

• In this specific case, it leads to
• \(81a^2 - 16\)

which efficiently showcases the power of the difference of squares technique.

Polynomial Identities

Polynomial identities are essential tools in algebra for transforming expressions. They allow for the simplification or expansion of polynomials.
These identities encapsulate patterns that frequently appear with polynomials. One such pattern is the difference of squares, which is just one in a list of useful identities like

• Perfect square trinomials
• Sum of cubes
• Difference of cubes

These identities are shortcuts, enabling you to bypass some traditional procedural steps.
By using polynomial identities, particularly in exercises involving binomials like

• \((9a-4)(9a+4)\),

we noticed an efficient way to arrive at

• \(81a^2 - 16\).

This not only reduces the computational workload but also makes understanding algebraic expressions more intuitive. Recognizing and applying these identities are fundamental for mastering polynomial manipulations.

Algebraic Expressions

Algebraic expressions form the core of algebra, consisting of variables, constants, and operations (like addition and subtraction). Understanding these expressions is crucial for solving equations and simplifying expressions. When dealing with algebraic expressions like

• \((9a-4)(9a+4)\),

it's often helpful to classify the type of expression you're working with.
Expressions are structured meaningfully:

• Terms are combined using operations
• Coefficients tell how many times a term is multiplied
• Variables stand for numbers that can change

Through logical manipulation of these elements, you can shift from one form to another, often simplifying the expressions or setting up for solving an equation.
The given exercise used all these components, transforming the expression

• from binomials into a simplified polynomial:
• \(81a^2 - 16\)

Understanding the structure and relationships within algebraic expressions is key to progressing in algebra.