Problem 100 - Algebra A Combined Function

Question

Write a perfect square trinomial that factors as \((x+3 y)^{2}\)

Step-by-Step Solution

Answer

The perfect square trinomial is \(x^2 + 6xy + 9y^2\).

Step 1: Identify the Formula for a Perfect Square Trinomial

A perfect square trinomial that factors as \((a + b)^2\) has the form \(a^2 + 2ab + b^2\). This is the expanded form of the square of a binomial.

Step 2: Identify 'a' and 'b'

In the expression \((x+3y)^2\), identify \(a\) and \(b\) as follows: \(a = x\) and \(b = 3y\). These are the terms that make up the binomial.

Step 3: Square 'a' to Form the First Term

Calculate \(a^2\):\[a^2 = x^2\]

Step 4: Square 'b' to Form the Third Term

Calculate \(b^2\):\[b^2 = (3y)^2 = 9y^2\]

Step 5: Calculate the Middle Term

The middle term of a perfect square trinomial is given by \(2ab\):\[2ab = 2(x)(3y) = 6xy\]

Step 6: Combine All the Terms

Combine \(a^2\), \(2ab\), and \(b^2\) to write the perfect square trinomial:\[x^2 + 6xy + 9y^2\]

Key Concepts

Binomial Expansion

In algebra, the binomial expansion is a technique used to expand expressions that involve binomials raised to a power. The classic example is the expansion of \[(a+b)^2\], which unfolds as a perfect square trinomial: \[a^2 + 2ab + b^2.\] This pattern comes from multiplying the binomial by itself and helps in understanding how each term arrives at its form.

• The first term is the square of the first term in the binomial, \(a^2\).
• The middle term is twice the product of the two terms, which is expressed as \(2ab\).
• The last term is the square of the second term, \(b^2\).

In the original exercise, we worked with \((x + 3y)^2\). Using the binomial expansion, we identify\(a = x\) and \(b = 3y\).By expanding this binomial, we form the perfect square trinomial \(x^2 + 6xy + 9y^2\). This example shows how binomials raised to powers can be efficiently expanded using this method.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra, involving rewriting expressions as products of simpler factors. For perfect square trinomials, you can commonly employ the formula derived from binomial expansion.
A perfect square trinomial follows the format \(a^2 + 2ab + b^2\) and can be factored back to the original binomial expression squared. In our exercise, \(x^2 + 6xy + 9y^2\) can be factored as \((x + 3y)^2\).
When factoring these trinomials, the process involves:

• Recognizing the first and last terms as squares: \(x^2\) involves \(x\), and \(9y^2\) involves \(3y\).
• Finding the middle term that fits the format: \(2ab = 6xy\), where \(a = x\) and \(b = 3y\).
• Confirming the structure matches \((a + b)^2\).

By efficiently rewriting the trinomial as a square of a binomial, you simplify expressions that might later be used in more complex equations.

Algebraic Expressions

In algebra, we deal with algebraic expressions—combinations of numbers, variables, and arithmetic operations like addition and multiplication. These expressions form the basis of more complex mathematical problems. Understanding how to manipulate them is crucial.
A perfect square trinomial such as \(x^2 + 6xy + 9y^2\) is an algebraic expression that can be broken down through processes like factoring and binomial expansion. It consists of:

• Variables: letters like \(x\) and \(y\), which represent unknowns or changeable values.
• Coefficient: the number multiplicatively associated with a variable, as in \(6xy\).
• Arithmetic operations: the inclusion of addition or multiplication within the expression, joining different terms.

Deciphering these expressions involves breaking them down into simpler parts, recognizing patterns like the perfect square trinomial, or choosing strategic paths for simplifying. These skills are essential as you progress with algebra and solve increasingly complex problems.