Problem 102 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 4 x^{3}+4 x^{2}+x $$

Step-by-Step Solution

Answer

The polynomial is factored as \(x(2x + 1)^2\).

Step 1: Identify the Greatest Common Factor (GCF)

Examine each term of the polynomial \(4x^3 + 4x^2 + x\) to determine if they share a common factor. Notice that each term has at least one \(x\) in it. Thus, the GCF is \(x\).

Step 2: Factor Out the GCF

Divide each term of the polynomial by the GCF \(x\) and factor it out, resulting in: \[x(4x^2 + 4x + 1)\]

Step 3: Factor the Quadratic Expression

Look at the quadratic \(4x^2 + 4x + 1\). We need two numbers that multiply to \(1 \times 4 = 4\) and add to \(4\). Notice that \((2, 2)\) satisfies this condition. We rewrite the quadratic as follows: \[4x^2 + 2x + 2x + 1\].

Step 4: Group and Factor by Grouping

Group the terms as \((4x^2 + 2x) + (2x + 1)\). Factor each group: \[2x(2x + 1) + 1(2x + 1)\].

Step 5: Factor out the Common Binomial Factor

Since \(2x + 1\) is a common factor in both groups, factor it out: \[(2x + 1)(2x + 1)\].

Step 6: Combine All Factorizations

Combine the initial factor with the factorization from Steps 4 and 5, resulting in the fully factored form of the polynomial: \[x(2x + 1)^2\].

Key Concepts

Greatest Common Factor (GCF)

The first step in factoring polynomials is to identify the Greatest Common Factor (GCF). This involves finding the largest expression that evenly divides each term in the polynomial.
In our given polynomial, each term is examined for common factors. The terms are

• \(4x^3\)
• \(4x^2\)
• \(x\)

Notice that all terms have at least one \(x\) as a factor. Since there is no numerical GCF aside from 1, \(x\) is the GCF.
This concept is crucial because factoring out the GCF simplifies the polynomial, making the remaining expression easier to handle in further steps.

Factoring Quadratics

After factoring out the GCF, the polynomial may simplify into a quadratic expression. Recognizing and factoring quadratics is a fundamental skill in algebra.
The quadratic expression in our example is: \[4x^2 + 4x + 1\] To factor it, we are looking for two numbers that multiply to the product of the quadratic coefficient (4) and the constant term (1), which is 4, and add to the middle coefficient, which is also 4.
In this case, the numbers 2 and 2 fit perfectly. Utilizing these numbers, rewrite the middle term: \[4x^2 + 2x + 2x + 1\]This allows us to set up the expression for grouping, paving the way for the next step in factoring.

Binomial Factorization

Binomial factorization involves grouping terms and identifying common binomial expressions. Once the quadratic is reorganized, a crucial step follows: factor by grouping.
Take the rearranged quadratic: \[(4x^2 + 2x) + (2x + 1)\]Group the terms and factor each separately:

• First group: \(2x(2x + 1)\)
• Second group: \(1(2x + 1)\)

Here, both groups share a binomial factor: \(2x + 1\). This "common factor" can be factored out as: \[(2x + 1)(2x + 1)\] or \((2x + 1)^2\).The initial factor \(x\) is then combined with these results to give the final fully factored form: \[x(2x + 1)^2\]. This highlights the power of binomial factorization in simplifying and solving polynomial expressions.