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Chapter 5: Problem 94

Factor each polynomial completely. $$-2 m^{2} n-8 m n-8 n$$

Short Answer

Expert verified
-2n(m + 2)^2

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Find the GCF of the terms \( -2m^2n, -8mn, \text{and} -8n \). The common factor among the coefficients \( -2, -8, -8 \) is \( -2 \). Additionally, the common variable factor is \( n \). Therefore, the GCF of the polynomial is \( -2n \).
02

Factor out the GCF

Factor out \( -2n \) from each term of the polynomial. This gives: \[ -2m^2n - 8mn - 8n = -2n(m^2 + 4m + 4). \]
03

Factor the Remaining Polynomial

Factor the quadratic polynomial \( m^2 + 4m + 4 \). Notice that it is a perfect square trinomial: \[ m^2 + 4m + 4 = (m + 2)^2 \].
04

Combine the Factors

Combine the GCF with the factored form of the quadratic polynomial to get the completely factored expression: \[ -2n(m + 2)^2 \].

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Key Concepts

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

Greatest Common Factor (GCF)
When factoring polynomials, the first step often involves finding the Greatest Common Factor (GCF). The GCF is the largest factor that divides all the terms in the polynomial. To find the GCF, you look at both the numerical coefficients and the variables.

In the given exercise, the polynomial is e -2m^{2}n - 8mn - 8n . The coefficients are -2, -8, and -8; their GCF is -2. All terms also have the variable n in common, so the overall GCF is -2n.

When you factor out the GCF, the original polynomial is rewritten as: -2n(m^2 + 4m + 4) .

This makes the polynomial simpler and easier to work with.
Perfect Square Trinomial
A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. To identify a perfect square trinomial, check if the first and last terms are perfect squares, and the middle term is twice the product of their square roots.

In the exercise, after factoring out the GCF, we are left with
m^2 + 4m + 4 . Notice that m^2 is a perfect square since it is m * m , and 4 is a perfect square since it is 2 * 2. The middle term 4m is twice the product of m and 2. Therefore, m^2 + 4m + 4 can be factored as: (m + 2)^2 .

Recognizing perfect square trinomials helps in simplifying polynomials quickly.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2. It has the general form: ax^2 + bx + c , where a, b, and c are constants. Factoring quadratic polynomials is a common method used to simplify and solve quadratic equations.

In our example, after removing the GCF and simplifying, the quadratic polynomial is: m^2 + 4m + 4 . This quadratic can be factored further by recognizing it as a perfect square trinomial: (m + 2)^2 .

Combining this with the GCF factored out earlier, the fully factored form of the original polynomial is: -2n(m + 2)^2 .

Mastering the factoring of quadratic polynomials is essential for solving a wide range of algebraic problems smoothly.

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Most popular questions from this chapter

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