Problem 73 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 25 w^{3}-10 w^{2}+w $$

Step-by-Step Solution

Answer

The completely factored form is \(w(5w-1)^2\).

Step 1: Factor Out the Greatest Common Factor (GCF)

Identify the greatest common factor in the terms. The terms are: \(25w^3\), \(-10w^2\), and \(w\). The common factor is \(w\). Factor \(w\) out of each term. \[25w^3 - 10w^2 + w = w(25w^2 - 10w + 1)\]

Step 2: Factor the Quadratic Expression

Factor the quadratic expression inside the parentheses: \(25w^2 - 10w + 1\). Attempt to factor by grouping or through the use of the quadratic formula. Identify two numbers that multiply to \(25 \times 1 = 25\) and add to \(-10\). These numbers are \(-5\) and \(-5\).\[25w^2 - 10w + 1 = (5w - 1)(5w - 1)\]

Step 3: Combine the Factors

Write the expression as a product of all identified factors. Combine the GCF and the factored quadratic.\[25w^3 - 10w^2 + w = w(5w - 1)^2\]

Step 4: Identify Any Prime Polynomials

Check if any further factorization is possible. The quadratic factors are already in their simplest form, meaning the factors are prime. Conclude the factorization process since no further factorization is possible.

Key Concepts

Greatest Common Factor

To begin factoring a polynomial, we first look for the greatest common factor (GCF). The GCF is the largest factor that divides all terms of the polynomial. In the given polynomial \(25w^3 - 10w^2 + w\), we identify the common factor in each term.

• The terms are: \(25w^3\), \(-10w^2\), and \(w\).
• Each term has a common factor of \(w\).

By factoring \(w\) out of each term, we simplify the expression:
\(25w^3 - 10w^2 + w = w(25w^2 - 10w + 1)\).
This step makes it easier to address more complex portions of the polynomial.

Quadratic Expression

After factoring out the GCF, we focus on the quadratic expression inside the parentheses: \(25w^2 - 10w + 1\). Quadratic expressions are polynomials of degree 2, typically in the form \(ax^2 + bx + c\).
Here:

• \(a = 25\) (the coefficient of \(w^2\))
• \(b = -10\) (the coefficient of \(w\))
• \(c = 1\) (the constant term)

Factoring a quadratic expression typically involves finding two binomials whose product gives back the quadratic expression. We find factors that multiply to \(a \times c = 25 \times 1 = 25\) and add to \(b = -10\). The numbers \(-5\) and \(-5\) fit these conditions: \(25w^2 - 10w + 1 = (5w - 1)(5w - 1) = (5w - 1)^2\).

Prime Polynomial

A prime polynomial cannot be factored further using integer coefficients. In this problem, after we factor out GCF and rewrite the quadratic expression, we need to check if it’s a prime polynomial.
We did this by splitting \(25w^2 - 10w + 1\) into two binomials \((5w - 1)(5w - 1)\). Each binomial here is already in its simplest form, showcasing that the quadratic factors of the original polynomial are already prime.
No further factoring is possible, indicating that all the polynomials involved are prime in their respective form.

Factorization Steps

The steps needed to factorize any polynomial can simplify the process considerably:

• Step 1: Identify and factor out the greatest common factor (GCF).
• Step 2: Address any remaining quadratic expressions by factoring further.
• Step 3: Combine all factors identified in each step.
• Step 4: Check for prime polynomials to confirm if any further factorization is necessary.

Using these steps, we approach factorization methodically:

• We started with \(25w^3-10w^2+w\).
• Factored out \(w\) as GCF: \(25w^3 - 10w^2 + w = w(25w^2 - 10w + 1)\).
• Factored the quadratic: \(25w^2 - 10w + 1 = (5w - 1)(5w - 1)\).
• Combined all parts: \(w(5w - 1)^2\).

The result is an elegant and comprehensible factorization of the original polynomial.