Problem 62 - Introductory Algebra for College Students

Question

Factor completely. $$3 w^{4}+54 w^{3}+135 w^{2}$$

Step-by-Step Solution

Answer

So, the factored form of the given expression is \(3w^2(w + 3)(w + 15)\).

Step 1: Identifying the Greatest Common Factor (GCF)

Looking at the given expression \(3 w^{4}+54 w^{3}+135 w^{2}\), the GCF that is common to all terms appears to be \(3w^2\). We can therefore factor this out from each individual term in the polynomial.

Step 2: Factoring out the GCF

When we factor out \(3w^2\) we get \(3w^2(w^2 + 18w +45)\). Factoring out means dividing each term in the polynomial by the GCF.

Step 3: Further Simplifications

Next task is to factorize the quadratic expression \(w^2 + 18w + 45\). This can be expressed as two binomial products. For this, look for two numbers that multiply to \(45\) and add to \(18\). The numbers \(3\) and \(15\) fulfill these requirements. So, the expression can be rewritten as: \(3w^2(w + 3)(w + 15)\).

Key Concepts

Greatest Common Factor

A crucial step in factoring polynomial expressions is identifying the Greatest Common Factor (GCF). The GCF is the largest factor shared by all terms in a polynomial. Finding it allows for simplification and is the first step towards factoring the polynomial completely.
For example, in the expression \(3 w^{4}+54 w^{3}+135 w^{2}\), the GCF needs to be determined. First, examine the numerical coefficients: 3, 54, and 135. The common factor in these numbers is 3. Then, look at the variable components, \(w^4\), \(w^3\), and \(w^2\). The smallest exponent of \(w\) present in all terms is 2.

• Combine these elements to get the GCF: \(3w^2\).

Once the GCF is known, factor it out of the polynomial by dividing each term by \(3w^2\), simplifying the expression drastically. This step paves the way for further factoring processes as seen in the polynomial simplification.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving a sum of powers in one or more variables multiplied by coefficients. They look something like this: \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\). In our specific problem, \(3 w^{4}+54 w^{3}+135 w^{2}\) is a polynomial.
Identifying each term's degree in a polynomial is helpful. The degree is determined by the highest power of the variable in the term. Here, the polynomial has degrees 4, 3, and 2, respectively for each term.
When factoring polynomials, our goal often involves simplifying these expressions to uncover the roots or solutions of polynomial equations. By factoring, we rewrite the expression as a product of simpler polynomials, often in the form of binomials or trinomials.

Quadratic Expression

Once we've factored out the GCF, we're often left with a simpler expression. In this case, after factoring out \(3w^2\), we simplify the polynomial to \(w^2 + 18w + 45\), which is a quadratic expression. Quadratics are polynomials of degree 2 and can often be factored into products of binomials.
The quadratic can be expressed as \(a x^2 + bx + c\). Here, \(a=1\), \(b=18\), and \(c=45\). To factor this expression, look for two numbers that multiply to 45 (the constant term \(c\)) and add to 18 (the coefficient of \(b\)). These numbers are 3 and 15.

• By using these numbers, express the quadratic as \((w + 3)(w + 15)\).

Factoring quadratics is a fundamental process when simplifying polynomial expressions further and solving equations. With the quadratic expression factored, the original polynomial becomes easier to interpret and handle.