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Chapter 5: Problem 88

Factor each polynomial completely. $$2 b n^{2}-4 b^{2} n+2 b^{3}$$

Short Answer

Expert verified
\(2b(n - b)^{2}\)

Step by step solution

01

Identify common factors

Examine the polynomial \(2 b n^{2} - 4 b^{2} n + 2 b^{3}\). Identify the greatest common factor (GCF) of all the terms. The GCF in this case is \(2b\).
02

Factor out the GCF

Factor out \(2b\) from each term in the polynomial: \(2b(n^{2} - 2bn + b^{2})\).
03

Factor the quadratic expression

Observe the quadratic expression inside the parentheses: \(n^{2} - 2bn + b^{2}\). Notice that this quadratic can be factored into: \((n - b)^{2}\).
04

Write the final factored form

Combine the factored GCF with the factored quadratic to get the final answer: \(2b(n - b)^{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
When you need to factor a polynomial, the first step is to look for the Greatest Common Factor (GCF). The GCF is the largest factor that all terms in the polynomial share.

Consider the polynomial from the exercise: \(2bn^{2} - 4b^{2}n + 2b^{3}\). To find the GCF, we look at the coefficients and variables of each term.
  • For the coefficients (2, -4, 2), the greatest common factor is 2.
  • For the variables, each term has at least one 'b'. So, 'b' is part of the GCF.

Therefore, the GCF is \(2b\). Factoring the GCF out from the polynomial simplifies the expression and makes it easier to handle: \(2b(n^{2} - 2bn + b^{2})\).

Finding the GCF is a crucial step because it reduces the complexity and prepares the expression for further factoring.
Factoring Quadratics
Once you have factored out the GCF, the next step is to factor the remaining quadratic expression.

In our example, the quadratic expression is \(n^{2} - 2bn + b^{2}\). Factoring quadratics often involves looking for patterns or using methods like completing the square or the quadratic formula.

Notice that \(n^{2} - 2bn + b^{2}\) fits the pattern of a perfect square trinomial: \((a-b)^{2} = a^{2} - 2ab + b^{2}\). Here it matches with \(a = n\) and \(b = b\). So, \(n^{2} - 2bn + b^{2}\) can be factored into: \((n - b)^{2}\).

Recognizing such patterns can simplify factoring quadratics significantly, making the process smoother and more intuitive.
Factored Form
After factoring out the GCF and simplifying the quadratic expression, you combine these to express the polynomial in its fully factored form.

In our exercise, we factored out the GCF \(2b\) and then simplified the quadratic \(n^{2} - 2bn + b^{2}\) into \((n - b)^{2}\). Combining these results, we get: \(2b(n - b)^{2}\).

The factored form is useful because it reveals the roots of the polynomial and simplifies further algebraic manipulation.
  • It makes polynomial equations easier to solve.
  • It helps in graphing polynomial functions.
  • It provides insight into the behavior of the polynomial.


Mastering these steps ensures you can tackle any polynomial and understand its structure more deeply.

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