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Chapter 5: Problem 119

Factor each expression completely. $$-14 a^{6} b^{6}+49 a^{2} b^{3}-21 a b$$

Short Answer

Expert verified
-7 a b (-2 a^{5} b^{5}+7 a b^{2}-3)

Step by step solution

01

Identify the Common Factor

Examine the terms \( -14 a^{6} b^{6}\), \(49 a^{2} b^{3}\), and \(-21 a b\). The common factors for these three terms are -7, \(a\), and \(b\). It's also important note that -7 is a common factor because all the coefficients are divisible by -7.
02

Factor Out the Common Factor

Now we factor out the common elements -7, \(a\), and \(b\). This yields \[ -7a b(-2 a^{5} b^{5}+7 a b^{2}-3 ) \]. So, the given expression factored out becomes \(-7 a b(-2 a^{5} b^{5}+7 a b^{2}-3 )\)
03

Verify the Factorization

To verify the correctness of your factorization, you can multiply the factors back together to see if you get the original polynomial. Multiply \(-7 a b\) by each term in the parenthesis \(-2 a^{5} b^{5}+7 a b^{2}-3 ). This gives the original polynomial, `-14 a^{6} b^{6}+49 a^{2} b^{3}-21 a b`, confirming that the factorization is correct.

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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 a common factor is central when factoring polynomials. A **common factor** is a number or expression that divides each term in a polynomial exactly without leaving a remainder. In our original exercise, we identified that -7, \(a\), and \(b\) are common factors. This means that these elements can be divided out from each term of the polynomial.

Finding a common factor involves looking at:
  • Coefficients: Check for numerical values that can divide each coefficient.
  • Variables: Look for variables that appear in each term and check their smallest exponent.
Identifying the common factor is crucial as it simplifies the expression and makes further factorization easier if needed.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and mathematical operators such as addition or subtraction. Each separate combination within an expression is known as a **term**. In our exercise, \(-14 a^{6} b^{6} + 49 a^{2} b^{3} - 21 ab\) is the algebraic expression presented for factorization.

An algebraic expression typically involves:
  • Variables: Symbols, often letters, that represent unknown values.
  • Coefficients: Numerical elements in front of variables.
  • Exponents: Indicate how many times a variable is multiplied by itself.
Understanding how these components interact in an expression is key to manipulating and simplifying expressions through factorization or other algebraic processes.
Polynomial Factorization
Polynomial factorization is the process of breaking down a polynomial into simpler components called **factors**. These factors, when multiplied together, give back the original polynomial. In our example, after identifying the common factor \(-7ab\), we simplify the expression to \(-7a b(-2 a^{5} b^{5}+7 a b^{2}-3 )\), which is the factored form of the polynomial.

The steps to factor a polynomial include:
  • Finding any common factors.
  • Rewriting the polynomial as a product of its factors.
  • Verifying the factorization by multiplying the factors to retrieve the original polynomial.
Factorization makes solving algebraic problems more convenient, as it reduces the polynomial to simpler, more manageable components. This technique is widely used in algebra to simplify expressions, solve equations, and understand polynomial behavior.

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Most popular questions from this chapter

Factor. $$p^{2}-2 p+1-q^{2}$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$-r^{2}+2 s^{2}+r s$$

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Explain how to determine whether a trinomial is prime.
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