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Chapter 6: Problem 101

What is factoring?

Short Answer

Expert verified
Factoring is a mathematical process where a composite number, expression or polynomial is expressed as a product of simpler numbers, expressions or polynomials. For instance, 48 can be factored into \(2 \times 2 \times 2 \times 2 \times 3\). The polynomial \(x^2 + 5x + 6\) can be factored into \((x + 2) (x + 3)\).

Step by step solution

01

Explaining the Concept of Factoring

Factoring is essentially decomposing a number, expression or a polynomial into a product of other numbers, expressions or polynomials. In general, if 'b' and 'c' are two factors of 'a', it means \(a = b \times c\).
02

Example of Factoring a Number

When we talk about factoring a number, we mean breaking it down into numbers that multiply together to give the original number. For example, the number 48 can be factored into \(2 \times 2 \times 2 \times 2 \times 3\). All these numbers multiplied together gives us the original number 48.
03

Example of Factoring a Polynomial

In the context of polynomials, factoring means expressing a polynomial as the product of its factors. For example, the polynomial \(x^2 + 5x + 6\) can be factored into \((x + 2)(x + 3)\). These two polynomials multiplied together gives us the original polynomial \(x^2 + 5x + 6\).

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Most popular questions from this chapter

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$12 y^{3}+16 y^{2}-3 y-4$$

Exercises 150鈥�152 will help you prepare for the material covered in the next section. Evaluate \((3 x-1)(x+2)\) for \(x=\frac{1}{3}\)

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$21 x^{2}-35 x y$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$3 x^{2}+243$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{2}-9=(x-3)^{2}\) for any real number \(x\)
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