36. What factor is common to all the expressions $x^2-x, 2 x^2+x-1$?
A. $x$
B. $x-1$
C. $x+1$
D. No common factor
Answer (1)
Factor the first expression: x 2 − x = x ( x − 1 ) .
Factor the second expression: 2 x 2 + x − 1 = ( 2 x − 1 ) ( x + 1 ) .
Compare the factors of both expressions.
Identify that there is no common factor: No common factor .
Explanation
Problem Analysis We are given two expressions, x 2 − x and 2 x 2 + x − 1 , and we need to find the common factor.
Factorizing the first expression First, let's factorize the expression x 2 − x . We can factor out an x from both terms: x 2 − x = x ( x − 1 )
Factorizing the second expression Next, let's factorize the expression 2 x 2 + x − 1 . We are looking for two numbers that multiply to 2 ∗ ( − 1 ) = − 2 and add up to 1 . These numbers are 2 and − 1 . So we can rewrite the middle term as 2 x − x :
2 x 2 + x − 1 = 2 x 2 + 2 x − x − 1 Now, we can factor by grouping: 2 x 2 + 2 x − x − 1 = 2 x ( x + 1 ) − 1 ( x + 1 ) = ( 2 x − 1 ) ( x + 1 )
Identifying common factors Now we have the factorized forms of both expressions: x 2 − x = x ( x − 1 ) 2 x 2 + x − 1 = ( 2 x − 1 ) ( x + 1 ) We can see that there are no common factors between the two expressions.
Final Answer Therefore, the answer is 'No common factor'.
Examples
Factoring expressions is a fundamental skill in algebra and is used extensively in various real-world applications. For instance, in engineering, factoring can help simplify complex equations that model physical systems, making them easier to analyze and solve. In economics, factoring can be used to analyze cost and revenue functions to determine break-even points and optimize profits. Moreover, in computer science, factoring is used in cryptography to secure data and communications. Understanding how to factor expressions allows us to simplify problems and find solutions more efficiently.
