Problem 84 Factor each expression completel... [FREE SOLUTION]

Chapter 6: Problem 84

Factor each expression completely. \(m^{2}(n+8)-9(n+8)\)

Short Answer

Expert verified

The expression is completely factored as \((n+8)(m+3)(m-3)\).

Step by step solution

Identify the common factor

Observe that the expression \(m^{2}(n+8)-9(n+8)\) has a common factor in both terms. In this case, \((n+8)\) is the common factor. Factoring out \((n+8)\) from the expression gives:\[(n+8)(m^{2} - 9)\]

Recognize the difference of squares

Notice that the expression \(m^{2} - 9\) is a difference of squares. Recall that a difference of squares can be expressed as \(a^2 - b^2 = (a+b)(a-b)\). Here, \(m^{2}\) is \(a^{2}\) and \(9\) is \(b^2\), where \(a = m\) and \(b = 3\).

Factor the difference of squares

Using the difference of squares formula, factor \(m^{2} - 9\) into \((m + 3)(m - 3)\). So, the factored form of \(m^{2} - 9\) is:\[(m + 3)(m - 3)\]

Write the completely factored expression

Combine the results from Steps 1 and 3. The completely factored form of the original expression is:\[(n+8)(m+3)(m-3)\]

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Key Concepts

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

Common Factor

The concept of finding a common factor plays a crucial role in simplifying algebraic expressions. When you have an expression like \(m^{2}(n+8)-9(n+8)\), strong observation skills will help you identify that both terms share a common component. In this case, it's \((n+8)\). Factoring out this common element can simplify the expression significantly.

Look at each term in the expression carefully.
Identify any repeated components across the terms.
Factor out the common component to simplify the expression.

For instance, after factoring out \((n+8)\), you convert the original expression into \((n+8)(m^{2} - 9)\). This step helps in breaking down the problem, making it easier to tackle.

Difference of Squares

Factorization often involves recognizing special patterns in the expression, such as the difference of squares. This concept is vital in algebra as it allows further simplification of terms like \(m^{2} - 9\).

A "difference of squares" is a specific type of expression that can be written as \(a^{2} - b^{2}\).
These expressions can be factored into the product of two binomials: \((a+b)(a-b)\).

Taking \(m^{2} - 9\) as an example, we can see it's made up of the squares \(m^{2}\) and \(3^{2}\). By applying the difference of squares formula, it factors to \((m+3)(m-3)\). Recognizing these structures is key to efficiently factoring polynomials.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial expression into a product of simpler polynomials. This skill is particularly useful for solving equations and simplifying expressions. The expression \(m^{2}(n+8)-9(n+8)\) showcases several steps in polynomial factoring:

Identify any common factors in the polynomial to simplify initially, as seen with \((n+8)\).
Look for special patterns, such as the difference of squares, to factor parts of the polynomial further.

The complete factoring process involves combining these simplified parts to form a fully factored expression, resulting in \((n+8)(m+3)(m-3)\). Mastery of polynomial factoring leads to greater ease in manipulating and solving complex algebraic expressions.

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91影视

Factor each expression completely. \(m^{2}(n+8)-9(n+8)\)

Short Answer

Step by step solution

Identify the common factor

Recognize the difference of squares

Factor the difference of squares

Write the completely factored expression

Key Concepts

Common Factor

Difference of Squares

Polynomial Factoring

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