Problem 21 - Elementary Algebra

Question

Factor. $$ 9 x 2-12 x y+4 y 2 $$

Step-by-Step Solution

Answer

The expression factors to \((3x - 2y)^2\).

Step 1: Identify a Perfect Square Trinomial

Notice the expression: \(9x^2 - 12xy + 4y^2\). This can potentially be a perfect square trinomial. A perfect square trinomial takes the form \(a^2 - 2ab + b^2\) and can be factored into \((a - b)^2\).

Step 2: Rearrange and Compare

Write the expression as is and compare it with the standard form of a perfect square trinomial. Expression: \(9x^2 - 12xy + 4y^2\)Compare to: \(a^2 - 2ab + b^2\)

Step 3: Identify 'a' and 'b'

From the expression, identify that \(a = 3x\) because \((3x)^2 = 9x^2\), and \(b = 2y\) because \((2y)^2 = 4y^2\). Ensure that the middle term works, as \(-2ab = -12xy\).

Step 4: Write the Factored Form

Since the expression matches the form \(a^2 - 2ab + b^2\), it can be factored as:\((3x - 2y)^2\).

Step 5: Verify the Solution

Verify by expanding \((3x - 2y)^2\) to check if it results in the original expression. Expand:\((3x - 2y)(3x - 2y) = 9x^2 - 6xy - 6xy + 4y^2 = 9x^2 - 12xy + 4y^2\).Since the expanded form matches the original expression, the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that results from squaring a binomial. To identify a perfect square trinomial, look for a polynomial of the form \[a^2 + 2ab + b^2\] or \[a^2 - 2ab + b^2\]. These can be factored into \((a + b)^2\) or \((a - b)^2\), respectively. This pattern arises because when a binomial is squared, the middle term (the \(2ab\) or \(-2ab\)) is twice the product of the binomial's terms. Identifying the perfect square trinomial makes it easier to factor complex expressions quickly. In our example, the polynomial \(9x^2 - 12xy + 4y^2\) fits this form, leading to the factored result \((3x - 2y)^2\). Recognizing these patterns can save time and simplify factoring tasks. It's like spotting a familiar face in math problems.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations that together form a mathematical phrase. Such expressions can include:

• constants (like 3 or 5),
• variables (like \(x\) or \(y\)),
• coefficients (numbers multiplying variables, like 9 in \(9x^2\)), and
• operators (such as +, -, \(\times\), and \(\div\)).

Understanding how to manipulate these components is key in solving equations and simplifying expressions. In our trinomial, \(9x^2 - 12xy + 4y^2\), the expression combines quadratic terms. To factor such expressions, identifying similar structures helps in rewriting them in a simpler form or finding their factors. Mastering algebraic expressions allows students to solve a vast array of math problems and real-world situations, highlighting the power of algebra.

Binomial Expansion

The concept of binomial expansion is essential in understanding how products of binomials unfold into polynomials. When we expand a binomial square, such as \((a - b)^2\), it results in \(a^2 - 2ab + b^2\). This results from the distributive property applied twice:

• First, multiply each term in the first binomial by each term in the second.
• Combine like terms to form the expanded expression.

For example, expanding \((3x - 2y)^2\) involves:

• Multiplying \(3x\) by \(3x\) and \(-2y\),
• then \(-2y\) by \(3x\) and \(-2y\).

This results in \(9x^2 - 6xy - 6xy + 4y^2\). By combining like terms, we get the expanded polynomial, \(9x^2 - 12xy + 4y^2\). Being comfortable with binomial expansion enables students to reverse the process—factoring—and apply it to solve algebraic challenges efficiently.