Problem 82 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ 16 x^{2}+24 x y+9 y^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the perfect square trinomial is \((4x + 3y)(4x + 3y)\).

Step 1: Identify the trinomial as a perfect square trinomial

To see if the given trinomial is a perfect square trinomial, we need to check if it follows the general form \(a^2x^2 + 2abxy + b^2y^2\). The given trinomial is: $$ 16x^2 + 24xy + 9y^2 $$ We can see that \(16x^2 = (4x)^2\), \(9y^2 = (3y)^2\), and \(24xy = 2(4x)(3y)\). Therefore, it is a perfect square trinomial.

Step 2: Factor the trinomial

Now that we know the trinomial is a perfect square trinomial, we can factor it using the general formula \((ax + by)^2 = (ax + by)(ax + by)\). In this case, \(a=4x\) and \(b=3y\). Thus, we have: $$ (4x + 3y)^2 $$ So the factored form of the given trinomial is: $$ (4x + 3y)(4x + 3y) $$

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that can be factored into the square of a binomial. It means when expanded, a binomial squared results in a perfect square trinomial. This process involves recognizing two crucial forms:

• \(a^2x^2\)
• \(2abxy\)
• \(b^2y^2\)

These match the general pattern of a perfect square trinomial \(a^2x^2 + 2abxy + b^2y^2 = (ax + by)^2\). To identify a perfect square trinomial, confirm that:

• The first term is a perfect square (e.g., \(16x^2 = (4x)^2\)).
• The last term is a perfect square (e.g., \(9y^2 = (3y)^2\)).
• The middle term is twice the product of the square roots of the first and last terms (e.g., \(24xy = 2 \times 4x \times 3y\)).

When all these conditions align, you can confidently declare the trinomial as a perfect square trinomial.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They form the backbone of algebra and allow you to express relationships and perform operations symbolically.
For instance, the expression \(16x^2 + 24xy + 9y^2\) includes variables \(x\) and \(y\), with coefficients such as 16, 24, and 9. The operations consist of addition, multiplication, and squaring.
Understanding how to manipulate these expressions is a key skill. They allow us to translate real-world problems into a format that can be solved and understood using mathematical rules. Recognizing patterns in such expressions, like perfect square trinomials, helps simplify and solve equations more efficiently.

Factoring Techniques

Factoring is a technique used to simplify algebraic expressions by breaking them into products of simpler ones. This can make solving equations or understanding the relationships within the expression more manageable.
When dealing with polynomials, there are many methods to choose from, such as:

• Factoring out the Greatest Common Factor (GCF)
• Factoring by grouping
• Using special products like difference of squares and perfect square trinomials
• Using the ac method for trinomials where other patterns aren't obvious

In the case of perfect square trinomials, we're using a specific factoring technique that leverages the identity \((ax + by)^2 = a^2x^2 + 2abxy + b^2y^2\). Once identified, you can directly factor the expression into repeated binomials, simplifying the original polynomial by showing it as a squared binomial.