Problem 50 - Elementary Algebra

Question

For the following problems, factor the trinomials if possible. $$ 12 b^{4}+30 b^{2} a+12 a^{2} $$

Step-by-Step Solution

Answer

Answer: The factored form of the trinomial is \(6(b^2 + a)(2b^2 + 2a)\).

Step 1: Identify the common factors

Observe the trinomial \(12b^4 + 30b^2a + 12a^2\). The coefficients are 12, 30, and 12. The greatest common factor of these coefficients is 6. Let's factor out this common factor from the trinomial. $$ 6(2b^4 + 5b^2a + 2a^2) $$

Step 2: Observe the trinomial

Now, let's observe the trinomial \(2b^4 + 5b^2a + 2a^2\). Notice that it resembles a quadratic trinomial in the form of \(ax^2 + bx + c\). We can treat \(b^2\) as x, so it becomes \(2x^2 + 5ax + 2a^2\).

Step 3: Factor the quadratic trinomial

To factor the quadratic trinomial \(2x^2 + 5ax + 2a^2\), look for two binomials whose product is the trinomial. The factors of 2, which is the coefficient of \(x^2\), are (1, 2). The factors are (2a, 1) for 2\(a^2\). Based on these factors, we can form two binomials: $$ (x + a)(2x + 2a) $$ which multiply together, resulting in: $$ 2x^2 + 2ax + ax + 2a^2 = 2x^2 + 5ax + 2a^2 $$

Step 4: Replace x with \(b^2\)

Now that we have factored the quadratic trinomial, replace x with \(b^2\) to get the factorized form of the original trinomial: $$ (b^2 + a)(2b^2 + 2a) $$

Step 5: Write the final factored form

Don't forget to include the common factor that we factored out in Step 1. The final factored form of the given trinomial is: $$ 6(b^2 + a)(2b^2 + 2a) $$

Key Concepts

Quadratic Trinomials

A quadratic trinomial is an algebraic expression of the form \(ax^2 + bx + c\). This means it is the sum or difference of three terms, where the highest degree is 2, making it a quadratic. In the context of factoring, the goal is to express this trinomial as a product of two binomials. For example, in the problem given, once we factored out the common factor, we had the expression \(2b^4 + 5b^2a + 2a^2\).
This expression is quadratic in form where each power is a multiple of 2. What's unique here is that the variable \(b^2\) acts like the single variable \(x\) in a standard quadratic trinomial. Thus, by treating \(b^2\) as a whole unit, you can factor it in a more simplified manner.​
Quadratic trinomials are structured so that they can usually be factored into simpler polynomials. Identifying it correctly is key to breaking it down into components that are easier to handle.

Common Factors

Recognizing common factors is a critical step in simplifying algebraic expressions and factoring trinomials. A common factor is any number that divides all coefficients of terms in the algebraic expression evenly. This simplifies the expression and makes further steps easier.
In the given problem \(12b^4 + 30b^2a + 12a^2\), the coefficients 12, 30, and 12 have a greatest common factor of 6. Dividing each term by this common factor simplifies it to \(2b^4 + 5b^2a + 2a^2\), making it more manageable for further factoring.

• Identify the largest number that divides all terms evenly. For 12, 30, 12, it is 6.
• Factor it out: Simplify each term by dividing by the common factor.
• Rewrite the expression: The new expression will be cleaner and sometimes reveals further factoring options.

Breaking down a problem starting with common factors reduces complexity significantly and helps in shedding light on next possible steps in solving it.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. They're the foundations for understanding and solving algebra problems. An expression could be as simple as \(x + 2\), or as complex as the trinomial we are handling here.
It's important to understand the components of an algebraic expression:

• Variables like \(a\) and \(b\): The symbols representing unknown values.
• Coefficients: Numbers multiplying the variables, such as 12, 30, and so on.
• Operations: Addition, subtraction, multiplication, and division to structure the expression.

When dealing with algebraic expressions, factoring comes into play to simplify these expressions or solve equations containing them. This process of breaking down an expression makes it easier to handle and can solve real-world problems efficiently. Understanding the syntax and structure of algebraic expressions like the trinomials helps in streamlining the problem-solving process.