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Chapter 5: Problem 78

Factor each polynomial completely. $$(x-2) x-(x-2) 5$$

Short Answer

Expert verified
(x-2)(x-5)

Step by step solution

01

Identify the common factor

First, observe that both terms in the polynomial (x-2)x - (x-2)5 have a common factor of (x-2).
02

Factor out the common factor

Factor out the common factor (x-2) from both terms. This gives (x-2)(x) - (x-2)(5).
03

Apply the distributive property

Use the distributive property to combine the factored terms: (x-2)(x-5).
04

Write the final factored form

The polynomial in its completely factored form is (x-2)(x-5).

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Key Concepts

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

Common Factor
In algebra, finding a common factor helps simplify expressions and solve equations. A common factor is a term that appears in each part of the expression. Identifying it is the first step in factoring polynomials.

Let's look at the original problem: \( (x-2)x - (x-2)5 \) We see that both parts of the polynomial, \( (x-2)x \) and \( - (x-2)5 \), share \( (x-2)\) as a common factor. By recognizing this, we can start simplifying the polynomial more easily.

When searching for common factors:
  • Look for terms that are present in each expression.
  • This could be numbers, variables, or even expressions like (x-2).
Spotting that (x-2) is a common factor here is crucial. Without this, further steps would be quite complex.
Distributive Property
The distributive property is a key algebraic principle that makes factoring and expanding expressions simpler. It states that \(a(b + c )= ab + ac\).

In our exercise, after identifying the common factor, we factored it out: \( \text{ (x-2) } \). This left us with: \[ (x-2)(x) - (x-2)(5)\]
Here鈥檚 where the distributive property comes into play. We use it to combine terms inside the parentheses:
  • (x-2) is distributed over (x-5).
  • Essentially, we combine all terms under a single (x-2).
By applying the distributive property: \[ (x-2)(x - 5) \] we further simplify the expression. This step showcases the power of distribution in combining and simplifying terms.
Factored Form
Factored form is the simplified version of the polynomial where it's broken down into the product of its factors. The last step in our problem gives us the final factored form. Factoring makes solving equations easier and reveals the polynomial鈥檚 roots.

In the exercise, we recognized (x-2) as a common factor, then used the distributive property to combine terms. This left us with: \[ (x-2)(x-5) \]
Let鈥檚 break this down:
  • (x-2) and (x-5) are factors of the original polynomial.
  • Multiplying these factors will give us back the expanded form.
This factored form shows that our original polynomial \((x-2)x - (x-2)5\) can be expressed as \( (x-2)(x-5)\).

Understanding the factored form helps in various applications, like finding zeros of the polynomial or solving equations efficiently.

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