Problem 48 - 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. $$ 3 x^{3}+3 x^{2}-126 x $$

Step-by-Step Solution

Answer

The trinomial factors completely to \(3x(x - 6)(x + 7)\).

Step 1: Identify the GCF

The given trinomial is \(3x^3 + 3x^2 - 126x\). Look for the greatest common factor (GCF) present in all the terms. Here, all terms have a factor of 3 and \(x\). So, the GCF is \(3x\).

Step 2: Factor out the GCF

Now, factor out the GCF \(3x\) from the trinomial: \[3x(x^2 + x - 42)\]

Step 3: Factor the Quadratic Expression

Focus on the quadratic expression \(x^2 + x - 42\). We need to find two numbers that multiply to \(-42\) and add to \(1\). These numbers are \(7\) and \(-6\), because \(7 \times (-6) = -42\) and \(7 + (-6) = 1\).

Step 4: Factor the Quadratic Trinomial

Rewrite the middle term of the quadratic using the numbers found: \(x^2 + 7x - 6x - 42\).Group the terms: \((x^2 + 7x) + (-6x - 42)\).Factor by grouping: \[x(x + 7) - 6(x + 7)\].Notice \(x + 7\) is common:\((x - 6)(x + 7)\).

Step 5: Combine with the GCF

The expression factored completely, given the GCF, becomes:\[3x(x - 6)(x + 7)\].

Key Concepts

Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is a foundational step in factoring polynomials. It involves finding the largest factor that divides each term of the polynomial without a remainder. In the given trinomial \(3x^3 + 3x^2 - 126x\), we look at each term: \(3x^3\), \(3x^2\), and \(-126x\).

To find the GCF, check both the numerical coefficients (3 in this case) and the variables (\(x\)).
- All terms can be divided by \(3\).
- Every term has at least one \(x\).

Thus, the GCF is \(3x\). Extracting the GCF simplifies the expression, making subsequent steps easier. This reveals the basic structure of the polynomial, allowing you to focus on simpler parts. Factoring the GCF is crucial as it reduces the complexity, leaving us with a quadratic expression that can be further factored.

Quadratic Expressions

Quadratic expressions typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These expressions can often be factored into the product of two binomials. Main methods involve looking for two numbers that multiply to give \(ac\) and add to give \(b\).

For \(x^2 + x - 42\), the task is to find two numbers multiplying to \(-42\) (the product of \(a = 1\) and \(c = -42\)), while summing to \(1\) (the middle coefficient, \(b\)). Here, \(7\) and \(-6\) clearly fit these requirements because:
- \(7 \times (-6) = -42\)
- \(7 + (-6) = 1\)

This recognition allows transforming the quadratic expression into a form suitable for further factoring by grouping.

Factoring by Grouping

Factoring by grouping is a technique used to simplify the factorization of polynomials. It works by rearranging the polynomial terms into groups that make it easier to factor. In the quadratic expression \(x^2 + 7x - 6x - 42\), you begin by dividing into groups: \((x^2 + 7x)\) and \((-6x - 42)\).

Each group has a common factor:
- \(x(x + 7)\) is pulled out from the first group.
- \(-6(x + 7)\) is pulled out from the second group.

Now you can clearly see that \((x + 7)\) is a common factor between the groups. This lets you factor it out completely as \((x - 6)(x + 7)\). Ultimately, this method simplifies the equation, allowing you to handle even more complex expressions efficiently.