Problem 32 - Elementary and Intermediate Algebra

Question

Factor. $$ 9 x^{2}-48 x y+64 y^{2} $$

Step-by-Step Solution

Answer

The factored form is \((3x - 8y)^2\).

Step 1: Identify the Quadratic Expression

The given expression is a quadratic in two variables: \(9x^2 - 48xy + 64y^2\). This expression is in the standard form of a quadratic trinomial, \(ax^2 + bxy + cy^2\). Here, \(a = 9\), \(b = -48\), and \(c = 64\).

Step 2: Check if It's a Perfect Square Trinomial

A perfect square trinomial has the form \((px + qy)^2\). To check, calculate \(b^2 = (2p)(2q)\). For our expression: \[ 4pq = -48 \] Also, the conditions \(p^2 = a\) and \(q^2 = c\) need to hold true. Calculate \(\sqrt{a}\) and \(\sqrt{c}\): \(p = \sqrt{9} = 3\), \(q = \sqrt{64} = 8\).

Step 3: Verify the Middle Term

Verify if the middle term \(-48xy\) is indeed \(2pqxy\):Compute \(2pq = 2 \times 3 \times 8 = 48\). The middle term is \(-48xy\), which confirms the trinomial as a perfect square: \( -48 = -1 \times 48\). Thus, the expression is \((3x - 8y)^2\).

Step 4: Write the Factored Form

Since the trinomial is a perfect square, it can be rewritten in its factored form:\[ 9x^2 - 48xy + 64y^2 = (3x - 8y)^2 \]

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms, specifically of the form \( ax^2 + bxy + cy^2 \). Here, each term plays a specific role:

• The term \( ax^2 \) is the quadratic term in \( x \).
• The term \( bxy \) represents the linear interaction between \( x \) and \( y \).
• The term \( cy^2 \) is the quadratic term in \( y \).

Each of these terms can be identified by their respective coefficients \( a \), \( b \), and \( c \). In the expression \( 9x^2 - 48xy + 64y^2 \), we have:

• \( a = 9 \),
• \( b = -48 \),
• \( c = 64 \).

Recognizing this structure is the first step in determining the nature of the trinomial, which can guide us in factoring it effectively.

Perfect Square Trinomial

A perfect square trinomial takes the specific form \((px + qy)^2\). Identifying a quadratic trinomial as a perfect square involves checking particular conditions:

• First, the square roots of \( a \) and \( c \), which appear as \( p \) and \( q \) respectively, must multiply with each other, giving the middle term.
• In mathematical terms, \( 2pq = b \), where \( p = \sqrt{a} \) and \( q = \sqrt{c} \).

For our expression \( 9x^2 - 48xy + 64y^2 \):

• \( p = \sqrt{9} = 3 \),
• \( q = \sqrt{64} = 8 \).

Checking these, we find that:

• \( 2 \times 3 \times 8 = 48 \),

which confirms that the trinomial is a perfect square. This allows us to express it as \((3x - 8y)^2\). This conceptual recognition significantly simplifies the factoring process.

Factoring Techniques

Factoring is the process of breaking down a polynomial into its component factors, making it easier to work with. For quadratic trinomials, the key is to recognize the structure and apply appropriate techniques. Here are steps to factor effectively:

• Identify if the trinomial is a perfect square. This is done by checking the middle term as explained in perfect square trinomials.
• Once confirmed, represent the trinomial as \((px + qy)^2\), where \( p \) and \( q \) are derived from \( a \) and \( c \) respectively.
• Verify the middle term is double the product of \( p \) and \( q \). If it matches, the expression is a perfect square, simplifying to \((3x - 8y)^2\), as in our example.

Using these techniques not only makes factoring straightforward but also deepens the understanding of polynomial structures.