Problem 47 - 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. $$ 6 x^{3}+54 x^{2}+120 x $$

Step-by-Step Solution

Answer

The trinomial factors completely as \(6x(x + 4)(x + 5)\).

Step 1: Identify the Greatest Common Factor (GCF)

To solve the trinomial, first look for the greatest common factor of all three terms. Here, the given trinomial is \(6x^3 + 54x^2 + 120x\). The coefficients are 6, 54, and 120. The greatest common factor of 6, 54, and 120 is 6. All terms also have at least one `x`, so the GCF of the trinomial is \(6x\).

Step 2: Factor out the Greatest Common Factor (GCF)

Now factor out the GCF \(6x\) from each term in the trinomial: \(6x^3 + 54x^2 + 120x = 6x(x^2 + 9x + 20)\). This reduces the canonical expression to a simpler trinomial \(x^2 + 9x + 20\).

Step 3: Factor the Remaining Trinomial

Next, factor the trinomial \(x^2 + 9x + 20\). Look for two numbers that multiply to the constant term (20) and add to the linear coefficient (9). The numbers 4 and 5 meet these criteria since \(4 imes 5 = 20\) and \(4 + 5 = 9\). Therefore, \(x^2 + 9x + 20 = (x + 4)(x + 5)\).

Step 4: Combine All Factors

Combine the factors from the previous steps: The factored form of the original trinomial \(6x^3 + 54x^2 + 120x\) is \(6x(x + 4)(x + 5)\).

Key Concepts

Greatest Common Factor

When working with trinomials, one of the first steps to simplify the expression is finding the Greatest Common Factor (GCF). The GCF is the largest number, including variables if applicable, that divides all terms of the polynomial without remainder.
For the trinomial \(6x^3 + 54x^2 + 120x\), the coefficients are 6, 54, and 120. You find the GCF of these by listing their factors:

• Factors of 6: 1, 2, 3, 6
• Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
• Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

By comparing these lists, the greatest common factor among these numbers is 6.
Additionally, each term in the polynomial also contains the variable \(x\). The term with the smallest power of \(x\) is \(x^1\), which means each term includes at least one \(x\). Therefore, the complete GCF for \(6x^3 + 54x^2 + 120x\) is \(6x\).

Factoring Polynomials

Once you have factored out the GCF in polynomial expressions, the next step is to tackle the remaining simpler polynomial. Factoring polynomials often involve finding two numbers that both sum to the middle term's coefficient and multiply to the constant term. The expression we simplified to was \(x^2 + 9x + 20\).
To factor \(x^2 + 9x + 20\), we look for two numbers whose product is 20 and sum is 9. These numbers are 4 and 5. Therefore, we write the polynomial as a product of two binomials:\[(x + 4)(x + 5)\]
This method, known as **factoring by grouping**, simplifies the initial trinomial by breaking down the middle term into two terms. From there, it combines them again but within a different, multipliable framework.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and the operations that connect them. In our example, \(6x^3 + 54x^2 + 120x\), each term is a part of an algebraic expression. Understanding how to factor these expressions relies on recognizing patterns and manipulating terms based on arithmetic rules.
When simplifying complex algebraic expressions, begin by identifying common elements, like the GCF, to reduce the overall complexity.

• Isolate parts of the expression that share similarities
• Factor expressions to their simplest forms
• Use algebraic identities to simplify calculations

Successfully factoring algebraic expressions allows for easier problem-solving, as it often leads to solutions that can be either evaluated or further manipulated depending on the problem's demands.