Problem 28 - Algebra A Combined Function

Question

Factor each trinomial by grouping. Exercises 9 through 12 are broken into parts to help you get started. $$ 3 x^{3}+8 x^{2}+4 x $$

Step-by-Step Solution

Answer

The trinomial factors as \(x(3x + 2)(x + 2)\).

Step 1: Identify Common Factor

Check if there is a common factor in all the terms of the trinomial \(3x^3 + 8x^2 + 4x\). The greatest common factor among these terms is \(x\).

Step 2: Factor Out the Common Factor

Factor out \(x\) from every term in the trinomial: \(3x^3 + 8x^2 + 4x = x(3x^2 + 8x + 4)\).

Step 3: Re-organize the Quadratic Expression

Look at the quadratic expression \(3x^2 + 8x + 4\) inside the parenthesis after factoring out \(x\). Notice that this expression can be rearranged to form a quadratic trinomial.

Step 4: Find Two Numbers for Grouping

Consider the form \(ax^2 + bx + c\) where \(a = 3\), \(b = 8\), and \(c = 4\). Find two numbers that multiply to \(3 \times 4 = 12\) (the product of \(a\) and \(c\)) and add up to \(8\). These numbers are \(6\) and \(2\), since \(6 \times 2 = 12\) and \(6 + 2 = 8\).

Step 5: Apply Grouping

Rewrite \(3x^2 + 8x + 4\) using the numbers found: \(3x^2 + 6x + 2x + 4\). Then group the terms: \((3x^2 + 6x) + (2x + 4)\).

Step 6: Factor Each Group

Factor out the common factor in each group: from \(3x^2 + 6x\), factor out \(3x\), giving \(3x(x + 2)\). From \(2x + 4\), factor out \(2\), giving \(2(x + 2)\). The expression now is \(3x(x + 2) + 2(x + 2)\).

Step 7: Factor Out the Binomial

Notice the common binomial \(x + 2\). Factor it out: \((3x + 2)(x + 2)\).

Step 8: Combine All Factors

Combine the factored form from Step 2 and Step 7: the fully factored form is \(x(3x + 2)(x + 2)\).

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential tool in factoring polynomials, especially trinomials. It's the largest factor that divides all terms in an expression evenly.
To find the GCF:

• List the factors for each term.
• Identify the common factors.
• Choose the largest common factor.

For our trinomial, \(3x^3 + 8x^2 + 4x\), the terms are \(3x^3\), \(8x^2\), \(4x\).
By listing out the factors of each, we see \(x\) is the GCF. With the GCF of \(x\), we can simplify the expression by factoring it out. This reduces the trinomial to a simpler form, streamlining the process of further factoring and solving the expression.

Factoring by Grouping

Factoring by grouping is a strategic method used to factor polynomials with four or more terms.
It's especially useful when a direct factorization isn't apparent. After factoring out the GCF, look for ways to rearrange and group the terms.
With our expression, \(3x^2 + 8x + 4\), we first find numbers that meet two conditions:

• They multiply to the product of the first and last coefficients (\(3 \times 4 = 12\)).
• They add up to the middle coefficient, which is \(8\).

These numbers are \(6\) and \(2\).
We rewrite the expression as \(3x^2 + 6x + 2x + 4\).
Next, we group it as \((3x^2 + 6x) + (2x + 4)\).
This sets us up to factor each group individually.

Quadratic Expressions

Quadratic expressions are polynomials in the form of \(ax^2 + bx + c\).
They appear frequently in algebraic problems and often need to be factored.
Factoring a quadratic expression involves finding two binomials whose product is the original expression.
In our case, after grouping, the quadratic expression is broken down into smaller, more manageable parts.
Each group is then factored separately to simplify the expression, aiding in easy identification and factoring of common binomials.
Working with quadratics requires recognition of patterns, such as perfect square trinomials or other identifiable forms that can speed up the factoring process.

Binomial Factorization

Binomial factorization is a straightforward yet powerful aspect of algebra that allows us to simplify complex expressions.
Once we have factored out the GCF and applied grouping to break down the quadratic expression, we often end up with a common binomial factor.
For our expression, after grouping and factoring, we have terms like \(3x(x + 2) + 2(x + 2)\).
Observe the repeated binomial term \((x + 2)\).
By factoring out the common binomial, our expression simplifies to \((3x + 2)(x + 2)\).
Combining this result with our initial GCF, we arrive at the fully factored form \(x(3x + 2)(x + 2)\).
This simplification makes evaluating or solving polynomial equations much easier.