Problem 36 - Elementary and Intermediate Algebra

Question

Factor. $$ 4 x^{4}-20 x^{3}+25 x^{2} $$

Step-by-Step Solution

Answer

The factored form is \(x^2(2x - 5)^2\).

Step 1: Identify Common Factors

First, look for the greatest common factor (GCF) of all the terms in the polynomial. In the expression \(4x^4 - 20x^3 + 25x^2\), the common factor is \(x^2\). By factoring \(x^2\) out, we get: \(x^2(4x^2 - 20x + 25)\).

Step 2: Factor the Quadratic Expression

Next, we need to factor the quadratic \(4x^2 - 20x + 25\). Try to express it in the form \((ax - b)(cx - d)\) or consider completing the square.

Step 3: Trial Factoring by Grouping Protential Factors

Check potential factors of \(4x^2 - 20x + 25\) by attempting values for \(a\), \(b\), \(c\), and \(d\) such that the middle term (20x) is obtained when foiling. After testing possibilities, it can be rewritten as \((2x - 5)^2\).

Step 4: Write the Complete Factored Form

Substitute the factored quadratic into the expression: The expression \(4x^4 - 20x^3 + 25x^2\) factors as \(x^2(2x - 5)(2x - 5)\). Hence the complete factorization is \(x^2(2x - 5)^2\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a crucial concept in polynomial factorization. It refers to the largest factor that is common to all terms in a polynomial. Identifying the GCF is the first step when aiming to simplify or factor any algebraic expression. To find the GCF:

• List the factors for each term.
• Identify the common factors shared by all terms.
• Choose the largest common factor to factor out from the expression.

In our example problem, the polynomial is \(4x^4 - 20x^3 + 25x^2\). The terms \(4x^4\), \(20x^3\), and \(25x^2\) all have a common factor of \(x^2\). Upon factoring out \(x^2\), we simplify the expression to \(x^2(4x^2 - 20x + 25)\). This step not only makes the expression easier to work with but also sets the stage for further factorization.

Quadratic Expression

A quadratic expression is a polynomial of degree two, typically in the form \(ax^2 + bx + c\). Quadratics are common in algebra and come with various properties that make them unique among polynomials. Our task with a quadratic is often to factor it into products of two binomials, if possible. When dealing with the expression \(4x^2 - 20x + 25\) from the problem:

• Start by identifying potential forms like \((ax - b)(cx - d)\).
• Use techniques like trial and error or completing the square.

For this specific quadratic, it turns out to reveal a perfect square trinomial, breaking down into \((2x - 5)^2\). Recognizing that some quadratics are perfect square trinomials can greatly simplify factorization, as shown here where the expression simplifies neatly to a squared binomial.

Factoring by Grouping

Factoring by grouping is a helpful method for polynomials with four or more terms, but it also plays a role in testing potential factor pairings for quadratics when other methods seem complicated. This technique involves rearranging terms and identifying common factors within select groups of a polynomial. Here's how you use it:

• Group terms with common factors.
• Factor out these common factors from each group.
• Look for a common binomial factor and factor it out.

In the context of the quadratic \(4x^2 - 20x + 25\), while not purely solved by grouping, this approach leads to inspecting possible factor pairings by guessing and checking until we find that the quadratic can be expressed as \((2x - 5)^2\). Factoring by grouping highlights how different algebraic techniques can complement each other in the quest to break down polynomials into their simplest forms.