Problem 121 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(2 x^{4}-5 x^{3}-25 x^{2}\)

Step-by-Step Solution

Answer

The expression factors to \(x^2(x - 5)(2x + 5)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, look at the coefficients and the variable terms to identify the GCF of the expression. The coefficients 2, -5, and -25 do not have a numeric GCF greater than 1. However, all the terms contain at least an \(x^2\). So, factor out \(x^2\) from each term:\[2x^4 - 5x^3 - 25x^2 = x^2 (2x^2 - 5x - 25)\]

Step 2: Factor the Quadratic Expression

Next, focus on the quadratic expression \(2x^2 - 5x - 25\) within the parentheses. We need to factor this quadratic expression. Use the AC method where \(A=2\), \(B=-5\), and \(C=-25\). Calculate \(AC = 2 imes -25 = -50\). Look for two numbers that multiply to \(-50\) and add to \(-5\). These numbers are 5 and -10.

Step 3: Rewrite and Factor by Grouping

Rewrite the middle term \(-5x\) as \(+5x - 10x\):\[2x^2 - 5x - 25 = 2x^2 + 5x - 10x - 25\]Group the terms in pairs:\[(2x^2 + 5x) + (-10x - 25)\]Factor out the common factors in each group:\[x(2x + 5) - 5(2x + 5)\]Finally, factor out the common binomial factor \((2x + 5)\):\[(x - 5)(2x + 5)\]

Step 4: Write the Fully Factored Expression

Combine the factors, noting that the expression factored out in Step 1 was \(x^2\). The fully factored expression is:\[x^2(x - 5)(2x + 5)\]

Key Concepts

Quadratic Expressions

Quadratic expressions play a crucial role in algebra. They are polynomial expressions of degree 2, typically presented in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Understanding how to manipulate and factor them is key to solving a variety of mathematical problems.

In a quadratic expression, the term \( ax^2 \) is the "quadratic term," \( bx \) is the "linear term," and \( c \) is the "constant term." Several methods are available to factor these expressions, including completing the square, using the quadratic formula, and factorization based on patterns.

For the provided example, we utilize factorization techniques. In the context of our example \( 2x^2 - 5x - 25 \), the aim is to convert it into a product of two linear factors using methods such as the AC method. This involves multiplying \( a \) and \( c \) to find integers that add up to \( b \). This makes it possible to express the quadratic as a product of simpler terms.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that two or more numbers or terms have in common. It is used to simplify expressions, making factoring easier by identifying elements common to all terms.

In our example, the expression \( 2x^4 - 5x^3 - 25x^2 \) requires identifying the GCF of the polynomial's terms. The coefficients themselves (2, -5, -25) have no common divisors other than 1, but every term includes at least \( x^2 \).

By factoring \( x^2 \) out, we simplify the problem. This leaves \( x^2(2x^2 - 5x - 25) \), reducing the quadratic that remains to be factored further. Identifying and extracting the GCF is a fundamental step in making more complex algebraic expressions manageable.

Factoring by Grouping

Factoring by grouping is a strategic approach when dealing with certain quadratic expressions or polynomials of higher degrees. It involves rearranging terms and forming groups to find common factors.

For the quadratic expression \( 2x^2 - 5x - 25 \), the goal is to split the middle term using two numbers that multiply to \( AC = -50 \) (from multiplying \( A = 2 \) and \( C = -25 \)) and add to \( B = -5 \). In this example, we use 5 and -10 to rewrite \(-5x\) as \(5x - 10x\).

We then group the terms \((2x^2 + 5x)\) and \((-10x - 25)\) and factor each group individually, resulting in expressions where \( x(2x + 5) \) and \(-5(2x + 5) \) appear. Finally, the common binomial \((2x + 5)\) is factored out, culminating in the fully factored form \((x - 5)(2x + 5)\). This technique is a powerful tool for breaking down and simplifying polynomials.