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Chapter 5: Problem 45

Factor completely. \(4 a^{3}-49 a\)

Short Answer

Expert verified
The completely factored form is \(a (2a + 7)(2a - 7)\).

Step by step solution

01

Identify Common Factors

Look for any common factors in the terms of the expression. The expression is \(4a^3 - 49a\). Both terms have a common factor of \(a\).
02

Factor Out the Common Factor

Factor out the common factor from each term. The expression becomes: \[a (4a^2 - 49)\].
03

Recognize the Difference of Squares

The expression inside the parentheses, \(4a^2 - 49\), is a difference of squares. Recall that the difference of squares formula is \(x^2 - y^2 = (x+y)(x-y)\). Here, \(4a^2\) can be written as \((2a)^2\), and \(49\) can be written as \(7^2\).
04

Apply the Difference of Squares Formula

Factor \(4a^2 - 49\) using the difference of squares formula. We have: \[4a^2 - 49 = (2a)^2 - 7^2 = (2a + 7)(2a - 7)\].
05

Combine All Factors

Combine all the factors we have found. The complete factorization of the original expression is: \[4a^3 - 49a = a (2a + 7)(2a - 7)\].

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Key Concepts

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

Common Factors
When factoring polynomials, looking for common factors is a crucial first step. A common factor is a term that divides all parts of a polynomial evenly. For example, in the expression \(4a^3 - 49a\), both terms share the common factor \(a\).

To factor out the common factor, you take it out of each term in the polynomial. This leaves you with: \[4a^3 - 49a = a(4a^2 - 49)\].

Always check each term to ensure you have found all common factors. If they exist, factoring them out simplifies the polynomial and makes the next steps easier.
Difference of Squares
The difference of squares formula is a special factorization formula that applies to expressions of the form\(x^2 - y^2\). This expression can be factored into: \((x + y)(x - y)\).

For example, if you have \(4a^2 - 49\), it can be re-written as\((2a)^2 - 7^2\).

Using the difference of squares formula, you get: \((2a + 7)(2a - 7)\). This is an incredibly useful formula to recognize quickly, as it simplifies the factorization process significantly.
Polynomial Factorization
Polynomial factorization involves breaking down a polynomial into simpler expressions (factors) that, when multiplied together, give back the original polynomial. The previous steps are parts of the general process of polynomial factorization.

For the given example \(4a^3 - 49a\), we first factored out the common factor: \[a(4a^2 - 49)\].

Then, we recognized and applied the difference of squares rule: \[4a^2 - 49 = (2a + 7)(2a - 7)\]. Lastly, we combined the factors: \[4a^3 - 49a = a(2a + 7)(2a - 7)\].

Understanding these steps helps in solving more complex polynomial equations efficiently.

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Most popular questions from this chapter

Perform the indicated operations. $$ \left(x^{2}-4 x+7\right)+\left(3 x^{2}-9\right)-\left(x^{2}-4 x+7\right) $$

Factor completely. $$ 24 t^{2 a}-6 $$

Can the number of solutions of a quadratic equation exceed two? Why or why not?

Solve. A rectangular table is 60 in. long and 40 in. wide. A tablecloth that is twice the area of the table will be centered on the table. How far will the tablecloth hang down on each side?

Use a graphing calculator to find any real-number solutions that exist accurate to two decimal places. $$x^{3}-3.48 x^{2}+x=3.48$$
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