Problem 25 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ 9 x^{2}-24 x y+16 y^{2} $$

Step-by-Step Solution

Answer

The factored form of the trinomial is \((3x - 4y)^2\).

Step 1: Identify the trinomial structure

The given expression is a trinomial of quadratic form, \(9x^2 - 24xy + 16y^2\). We see terms involving \(x^2\), \(xy\), and \(y^2\), indicating we are dealing with a quadratic in terms of \(x\) and \(y\).

Step 2: Recognize the square form

Notice that each term is a perfect square or can be easily recognized as such. The expression resembles the form \((ax + by)^2\) where \(a = 3\) and \(b = -4\) because \((3x)^2 = 9x^2\), \((-4y)^2 = 16y^2\), and the middle term \(-24xy\) matches \(2*a*b\), specifically \(2(3)(-4)xy = -24xy\).

Step 3: Write as a perfect square

Based on the recognition from step 2, the trinomial \(9x^2 - 24xy + 16y^2\) can be expressed as a perfect square: \((3x - 4y)^2\). This is derived directly from understanding the square of a binomial formula: \((a + b)^2 = a^2 + 2ab + b^2\).

Step 4: Verify the factorization

To ensure the factorization is correct, expand \((3x - 4y)^2\): \((3x - 4y)(3x - 4y) = 9x^2 - 12xy - 12xy + 16y^2 = 9x^2 - 24xy + 16y^2\). This confirms the factorization is accurate.

Key Concepts

Quadratic Expressions

Quadratic expressions are algebraic expressions where the highest power of the variable is 2. In their general form, they look like this: \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Quadratic expressions can involve one or multiple variables, like in the exercise where both \( x \) and \( y \) are present. Quadratics are fundamental in algebra because they model a wide range of real-world phenomena.
Understanding how to handle them includes learning how to factor them, solve equations, and manipulate their forms to simplify problems. In the trinomial \( 9x^2 - 24xy + 16y^2 \), each term adheres to the quadratic format, making it a candidate for factoring using specific techniques.

• The first term, \( 9x^2 \), is a squared component related to the quadratic nature.
• The last term, \( 16y^2 \), similarly reflects another squared component.
• The middle term, \( -24xy \), connects these squares, contributing to the factorization process through recognizable patterns.

Recognizing this structure is a crucial step in simplifying and factoring quadratic expressions.

Perfect Square Trinomials

Perfect square trinomials are special quadratic trinomial expressions that can be rewritten as the square of a binomial. They follow a distinctive pattern: \( a^2 + 2ab + b^2 = (a + b)^2 \). Recognizing this pattern simplifies the process of factoring, as seen in the exercise with \( 9x^2 - 24xy + 16y^2 \).
In this expression:

• The term \( 9x^2 \) represents \( (3x)^2 \).
• The term \( 16y^2 \) is \( (4y)^2 \).
• The middle term, \( -24xy \), matches \( 2 imes (3x) imes (-4y) \), confirming the expression is \( (3x - 4y)^2 \).

Recognizing and simplifying perfect square trinomials is a powerful algebraic skill that opens doors to solving complex equations easily. It reduces the problem to manipulating simpler, square expressions and serves as a foundational tool in algebra.

Binomial Theorem

The binomial theorem is a remarkable principle that provides a formula for expanding binomials raised to any power. Its application isn't direct in basic polynomial factorization like perfect square trinomials, but understanding binomials is essential for recognizing patterns in trinomials.
In essence, when you have a binomial expression like \( (a + b)^n \), the binomial theorem gives a structured way to expand it, showing each term in terms of combinations and powers of \( a \) and \( b \). This is expressed as:\[ (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k \] where \( {n \choose k} \) is a binomial coefficient.

• In the case of smaller powers like \( (a + b)^2 \), it simplifies to the formula \( a^2 + 2ab + b^2 \), showing where the perfect square trinomial form is derived from.
• It aids significantly in predicting and constructing the forms of expressions analytically before solving or factoring them.

The critical takeaway for students is mastering the art of identifying when an expression can be handled with knowledge of binomials, helping secure accuracy and efficiency in mathematical problem-solving.