Problem 32 - Beginning and Intermediate Algebra

Question

Factor completely. $$100 n^{4}-8 n^{3}+64 n^{2}$$

Step-by-Step Solution

Answer

The completely factored expression is: \(4n^2(25n^2 - 2n + 16)\).

Step 1: Identify the greatest common factor (GCF)

Let's find the GCF of the terms in the given polynomial expression: \(100n^4\), \(-8n^3\), and \(64n^2\). The GCF of their coefficients is 4, and the GCF of their powers of n is \(n^2\). Therefore, the GCF of these terms is \(4n^2\).

Step 2: Factor out the GCF

We found the GCF to be \(4n^2\), so now let's factor that out from the given polynomial expression: \(4n^2(25n^2 - 2n + 16)\).

Step 3: Factor the remaining quadratic

Now we have to factor the remaining quadratic expression within the parentheses, \(25n^2 - 2n + 16\). In order to do this, we will use the general formula for factoring a quadratic expression of the form \(ax^2 + bx + c\): \((x + p)(x + q)\), where \(p\) and \(q\) are the roots. By finding the roots, we can determine the values of \(p\) and \(q\). However, upon attempting to find the roots of this quadratic using the quadratic formula or other methods, we find that it does not have any rational roots. Therefore, this quadratic is already factored completely.

Step 4: Write the final factored expression

Since the quadratic expression within the parentheses cannot be factored further, our final factored expression is: $$4n^2(25n^2 - 2n + 16)$$

Key Concepts

Greatest Common Factor

When tackling polynomials, one of the first steps in simplification is finding the **greatest common factor (GCF)**. This means identifying the largest factor that divides all the terms of a polynomial. Let's delve into what this means and how to find it effectively.Consider a polynomial with multiple terms, like the expression given, which is composed of terms such as \(100n^4\), \(-8n^3\), and \(64n^2\). Each of these terms consists of a numeric coefficient and a variable raised to a power.To find the GCF:

• Look at the numerical coefficients: 100, -8, and 64. The greatest common divisor for these numbers is **4**.
• Next, consider the variable part of each term, which in this case is powered by \(n\): \(n^4\), \(n^3\), and \(n^2\). The smallest power of \(n\) is \(n^2\), making it the GCF for the variable factors.

Putting it all together, the greatest common factor of the entire polynomial is **\(4n^2\)**. This factor serves as the greatest divisor common to all the terms, facilitating simplification by reducing the polynomial to a simpler form.

Quadratic Expression

A **quadratic expression** is one of the typical forms seen in algebra involving terms with a variable raised to the second power. It's usually written in the form \(ax^2+bx+c\), where \(a\), \(b\), and \(c\) are constants. In our given problem, once the GCF, \(4n^2\), is factored out from the polynomial, what's left is a quadratic expression, specifically \(25n^2 - 2n + 16\).Quadratics play a major role in algebra due to their predictive capabilities in graphing parabolas and solving real-world problems. To further factor a quadratic expression (if possible), various methods can be used:

• **Factoring by inspection**: If the roots are nice integers that can be determined by looking at the equation.
• **Quadratic formula**: Used for finding roots when factoring by inspection isn't feasible. This formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\) calculates the roots of the equation, but note, if the squared term under the root, called the discriminant \((b^2 - 4ac)\), is negative, the roots are not real numbers.

In this exercise, attempts to factor the quadratic further revealed that it cannot be broken down into simpler, rational components. This means that it is considered "factorable" in factoring terms, but only in the real number system as presented.

Factored Expression

A **factored expression** is the simplified version of an algebraic expression where common factors or terms are pulled out, revealing a product of terms. The goal of factoring is to reduce complexity. In the context of our polynomial, the factoring process started by extracting the greatest common factor, revealing a simpler quadratic.The process involves rewriting the polynomial expression in a multiplicative form highlighting its structure, which helps identify properties like roots and solutions easily. For our example, starting with the polynomial \(100n^4 - 8n^3 + 64n^2\), the factoring process leads us to:

• First remove \(4n^2\), the GCF, from each term, which leaves \((25n^2 - 2n + 16)\) within the parentheses.
• The complete expression now cannot be decomposed further in the real number space, and is thus considered fully factored as \(4n^2(25n^2 - 2n + 16)\).

This final form makes it clearer to understand, analyze, and graph the behavior of the function. Though it seems simple in presentation, this stage of expression can unveil deeper insights when applying algebraic principles in practical scenarios.