Problem 62 - Algebra A Combined Function

Question

Factor each trinomial completely. Some of these trinomials contain a greatest common factor (other than 1). Don't forget to factor out the GCF first. $$ 7 a^{3} b-35 a^{2} b^{2}+42 a b^{3} $$

Step-by-Step Solution

Answer

The trinomial factorizes to \(7ab(a - 2b)(a - 3b)\).

Step 1: Identify the Greatest Common Factor (GCF)

To begin factoring the trinomial \(7a^3b - 35a^2b^2 + 42ab^3\), first identify the greatest common factor (GCF) of all terms. \(7a^3b, -35a^2b^2,\) and \(42ab^3\) all have factors of 7, \(a\), and \(b\) in common.The GCF is \(7ab\).

Step 2: Factor Out the GCF

Once the GCF is identified, factor it out from each term in the polynomial:\[7a^3b - 35a^2b^2 + 42ab^3 = 7ab(a^2 - 5ab + 6b^2)\]This simplifies the polynomial, making it easier to factor further, if possible.

Step 3: Factor the Remaining Trinomial

Now focus on the trinomial inside the parentheses: \(a^2 - 5ab + 6b^2\). We need factors of 6 (the constant term) that add up to -5 (the coefficient of the middle term). The factors of 6 that satisfy this are -2 and -3:\(a^2 - 5ab + 6b^2 = (a - 2b)(a - 3b)\).

Step 4: Write the Completely Factored Form

Combine the GCF with the factored trinomial from Step 3 to get the completely factored form of the original polynomial:\[7ab(a - 2b)(a - 3b)\].

Key Concepts

Greatest Common Factor

The greatest common factor, often abbreviated as GCF, is crucial when factoring polynomials. It refers to the largest term that can be evenly divided into each term of the polynomial.
By identifying and factoring out the GCF first, you simplify the expression, making further factoring steps more manageable.
Let's break it down using an example from the exercise:

• Consider the polynomial terms: \(7a^3b, -35a^2b^2, \text{and } 42ab^3\).
• First, find the common factors in numbers: 7, -35, and 42. All are divisible by 7.
• Next, identify the common variables and their smallest exponents: \(a\) and \(b\) each appear in every term, with the lowest powers being \(a\) and \(b\).

Therefore, the GCF for these terms is \(7ab\). By factoring the GCF from the polynomial, the expression simplifies to an easier form: \(7ab(a^2 - 5ab + 6b^2)\). This process makes it accessible for further manipulation and factorization.

Trinomial Factoring

Trinomial factoring involves breaking down a trinomial into two or more simpler factors. A trinomial is a polynomial with three terms. To factor a quadratic trinomial like \(a^2 - 5ab + 6b^2\), you follow these steps:

• Identify the structure: Our trinomial follows the pattern \(ax^2 + bx + c\).
• Look for two numbers that multiply to the constant term \(c\) (here, 6), and add up to the coefficient of the middle term \(b\) (here, -5).
• For this trinomial, -2 and -3 are the numbers that work: \((-2) \times (-3) = 6\) and \((-2) + (-3) = -5\).

Thus, the trinomial can be factored into \((a - 2b)(a - 3b)\).
Effortlessly breaking down trinomials this way aids in solving more complex equations that may seem daunting at first glance.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding these basics helps in performing operations like factoring. Consider the expression \(7a^3b - 35a^2b^2 + 42ab^3\):

• Terms: Each part of an expression separated by a plus or minus sign is a term. Here, we have three terms.
• Coefficients: These are numbers multiplying the variables, such as 7 in \(7a^3b\).
• Variables: Symbols that represent numbers, \(a\) and \(b\) in this expression.

These components underlie operations such as factorization. Understanding such elements aids in identifying patterns, simplifying expressions, and solving equations.
Once you grasp these, factoring polynomials can even be as intuitive as simple arithmetic!