Problem 31 - Elementary and Intermediate Algebra

Question

Factor. $$ 81 x^{2}-72 x y+16 y^{2} $$

Step-by-Step Solution

Answer

The factored form is \((9x - 4y)^2\).

Step 1: Recognize Patterns

The given expression is a trinomial: \(81x^2 - 72xy + 16y^2\). Notice that both 81 and 16 are perfect squares and the expression resembles the form of a perfect square trinomial \(a^2 - 2ab + b^2\). This invites an initial guess that the trinomial might be expressible as \((a - b)^2\).

Step 2: Identify 'a' and 'b'

Examine the expression to find \(a\) and \(b\) by comparing \(a^2 - 2ab + b^2\) to the given trinomial. Identify \(a^2 = 81x^2\) leading to \(a = 9x\), and \(b^2 = 16y^2\) leading to \(b = 4y\).

Step 3: Verify with Middle Term

Check if the middle term \(-72xy\) matches the form \(-2ab\). Compute \(-2 \times 9x \times 4y = -72xy\). Since this holds true, the expression fits the form \((9x - 4y)^2\).

Step 4: Write the Factored Form

Since the trinomial is confirmed to be a perfect square, write the factored form as \((9x - 4y)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. Typically, it takes the form:

• \(a^2 + 2ab + b^2\)
• \(a^2 - 2ab + b^2\)

The trinomial given in the problem, \(81x^2 - 72xy + 16y^2\), is a classic example fitting the second pattern. To recognize such trinomials:

• Identify the structure where the first and last terms are perfect squares.
• Ensure that the middle term is twice the product of the terms that are square rooted from the first and last terms.

This means for our trinomial, \(a = 9x\) (since \((9x)^2 = 81x^2\)) and \(b = 4y\) (since \((4y)^2 = 16y^2\)). Once these are identified, verify the middle term, which should equal \(-2ab\). This pattern recognition helps efficiently factor the expression into \((9x - 4y)^2\). Such factoring simplifies working with higher degree polynomials.

Algebraic Expressions

Algebraic expressions are math sentences made from numbers, variables, and operations. They do not include an equality sign, separating them from equations.

• An expression like \(81x^2 - 72xy + 16y^2\) involves variables \(x\) and \(y\), coefficients 81, 72, and 16, and operations such as addition and multiplication.
• Understanding algebraic expressions is crucial for manipulating them into simpler or more useful forms, such as factoring.

In the context of this problem, algebraic expressions help represent the relationship between numbers and variables, enabling subsequent operations like factoring. Recognizing the structure of a perfect square trinomial within the expression allows us to break it down and simplify into a more manageable form. It's important to carefully pay attention to the signs and coefficients, as they guide whether you're dealing with addition or subtraction, and their placement in the binomial square.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler components, called factors, that when multiplied together return the original polynomial. Factoring is used to simplify expressions and solve polynomial equations.

• For a trinomial, identify it as a perfect square or using other methods like factoring by grouping, if applicable.
• Factoring can help solve equations since it reveals roots and intercepts via simpler binomial forms.

In this exercise's example, factoring \(81x^2 - 72xy + 16y^2\) into \((9x - 4y)^2\) helps simplify polynomial handling. It's crucial for students to practice recognizing patterns and applying factorization techniques. Once proficient, students can apply such skills to solve a wide range of algebraic problems involving polynomial functions. This enables better comprehension of both the operations involved and the implications of these transformations in various mathematical contexts.