Problem 50 - Elementary Algebra

Question

Factor completely. $$ 2 a 3 b_{2}-12 a_{2} b+18 a $$

Step-by-Step Solution

Answer

Factor: \( 2a(ab - 3)^2 \)

Step 1: Identify Common Factors

We start by looking at each term to find what, if any, common factors exist across all terms. The given expression is:\[ 2a^3b^2 - 12a^2b + 18a \]We notice that each term has at least a factor of \( a \). With the coefficients, 2, 12, and 18 have a common factor of 2. Thus, we have identified that 2a is a common factor for all terms.

Step 2: Factor Out the Common Factor

Next, we factor out the common factor \( 2a \) from the expression. We divide each term by \( 2a \):\[ 2a(a^2b^2 - 6ab + 9) \]

Step 3: Recognize a Special Binomial

We look at the expression inside the parentheses: \( a^2b^2 - 6ab + 9 \). The last term, 9, is a perfect square, suggesting we might have a perfect square trinomial. Can this trinomial be factored as a perfect square? It corresponds to the pattern \((x-y)^2 = x^2 - 2xy + y^2\).

Step 4: Factor the Trinomial

Here, \( x = ab \) because the first term is \( (ab)^2 \), and \( y = 3 \) because 9 is \( 3^2 \). It fits the form:\[ (ab)^2 - 2\cdot(ab)\cdot3 + 3^2 = (ab - 3)^2 \]So we can factor it as a square of the binomial:\[ (ab - 3)^2 \]

Step 5: Recombine the Factored Expression

By substituting the factorization of the trinomial back into the expression, we get:\[ 2a(ab - 3)^2 \]. Thus, the original expression is completely factored as \( 2a(ab - 3)^2 \).

Key Concepts

Common Factor

Identifying a common factor is the first step in simplifying many polynomial expressions. A common factor is a number or variable that evenly divides every term in a polynomial. For the expression given in the problem,

• We identify that each term has the variable "a" as a factor.
• The coefficients 2, 12, and 18 share a numeric factor of 2.

This means 2a is a common factor across all terms. Factoring it out simplifies the expression and helps in further manipulation. Once factored, the expression becomes \[ 2a(a^2b^2 - 6ab + 9) \]. Recognizing and extracting common factors like this can significantly simplify complex problems in algebra.

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial that can be rewritten as the square of a binomial. This form is crucial when factoring because it represents expressions such as \[(x-y)^2 = x^2 - 2xy + y^2\]. Consider the polynomial \[a^2b^2 - 6ab + 9\] from our factored expression:

• The first term, \(a^2b^2\), is the square of \((ab)\).
• The last term, 9, is the square of 3.
• The middle term, \(-6ab\), is twice the product of \(ab\) and 3.

These highlights fit the perfect square trinomial pattern, allowing us to factor it neatly into \((ab-3)^2\). Recognizing such patterns simplifies polynomial expressions and aids efficient problem-solving.

Binomial

A binomial comprises exactly two terms. In our context, when we note the factorization \((ab - 3)^2\), it's clear that \((ab - 3)\) itself is a binomial. Binomials serve as building blocks in algebraic expressions and can often be structured into other forms, like trinomials, by squaring or applying operations.

Understanding binomials is essential for managing more complex polynomial factorizations and recognizing patterns, such as perfect square trinomials, which rely on binomial structures for simplification. It's a robust toolset allowing you to dissect and handle algebraic expressions with agility.

Trinomial

A trinomial is a polynomial with three terms, much like the \(a^2b^2 - 6ab + 9\) encountered in our task. Breaking down trinomials is a rewarding problem-solving strategy as it often leads to their simplification into product forms of binomials.

• The leading term often suggests a squared variable or product, such as \(a^2b^2\).
• The middle term, here \(-6ab\), typically connects the first and last terms through multiplication factors.
• The last term is often a square or a straightforward multiplication result.

By understanding how a trinomial fits the form of a perfect square, in this case allowing \(a^2b^2 - 6ab + 9\) to become \((ab - 3)^2\), you uncover a clear path to simplified and neatly factored expressions.