Problem 60 - 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. $$ 3 x^{6}+30 x^{5}+72 x^{4} $$

Step-by-Step Solution

Answer

The completely factored form is \( 3x^4(x + 4)(x + 6) \).

Step 1: Identify the Greatest Common Factor (GCF)

First, observe the coefficients of each term: 3, 30, and 72. The GCF of these numbers is 3. Also, the smallest power of x (in all terms) is \( x^4 \). Therefore, the GCF of the entire expression is \( 3x^4 \).

Step 2: Factor out the GCF

Next, factor \( 3x^4 \) out of each term of the trinomial. This simplifies the expression as follows: \[ 3x^6 + 30x^5 + 72x^4 = 3x^4 (x^2 + 10x + 24) \].

Step 3: Factor the Trinomial

Now, we need to factor the quadratic trinomial \( x^2 + 10x + 24 \). Look for two numbers that multiply to 24 and add up to 10. These numbers are 4 and 6. Therefore, the trinomial can be rewritten as: \( x^2 + 10x + 24 = (x + 4)(x + 6) \).

Step 4: Combine Factors

Now that we've factored the quadratic trinomial, substitute it back into the expression factored by the GCF: \[ 3x^4(x + 4)(x + 6) \].

Key Concepts

Greatest Common Factor

When dealing with algebraic expressions, identifying the Greatest Common Factor (GCF) is a key step in simplifying the expression and preparing it for further factoring. The GCF is the largest factor that divides all the numbers in an expression. For numerical coefficients, this involves finding the largest number that divides each of the given numbers evenly.

For instance, if we look at the numbers 3, 30, and 72, we determine that their GCF is 3. However, the GCF also includes any common algebraic terms. In an expression like the one given in the exercise, each term also includes a power of x.

• Look at the smallest power of x in the expression for x terms.
• In the example, it's \( x^4 \).
• This means the overall GCF becomes \( 3x^4 \).

Once the GCF is determined, you can factor it out from the expression, simplifying your work.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, meaning that the highest power of the variable is 2. A typical quadratic trinomial has the form \( ax^2 + bx + c \). To factor these, we look for two numbers that satisfy two criteria:

• They must multiply to give \( c \).
• They must add to give \( b \).

In our exercise, after factoring out the GCF, we are left with the quadratic \( x^2 + 10x + 24 \). We find that the numbers 4 and 6 fit both criteria:

• They add up to 10.
• They multiply to 24.

This lets us rewrite the trinomial as \( (x + 4)(x + 6) \). Ensuring the correct pairing of these coefficients means the equation is completely factored.

Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations. When factoring them, like in our exercise, we aim to write the expression as a product of simpler expressions. This involves breaking down polynomials like trinomials into products of their factors.

The process begins by identifying any GCF, as we've discussed. But why do we factor? Factoring is essential because it simplifies expressions, making it easier to solve equations involving these expressions. It also reveals roots and aids in graphing functions. An expression like \( 3x^6 + 30x^5 + 72x^4 \) appears complex, but once factored, it becomes manageable: \( 3x^4(x + 4)(x + 6) \).

• Breaking down complex expressions into simpler ones is the essence of factoring.
• This main goal is to make solving and understanding easier.

Understanding these basics provides a solid foundation for more advanced topics in algebra.