36. What factor is common to all the expressions [tex]x^2-x[/tex], [tex]2 x^2+x-1[/tex] and
a. [tex]x[/tex]
C. [tex]x+1[/tex]
b. [tex]x-1[/tex]
d. No common factor

Factor x 2 − x to get x ( x − 1 ) .
Factor 2 x 2 + x − 1 to get ( 2 x − 1 ) ( x + 1 ) .
Compare the factors of both expressions and check the given options.
Conclude that there is no common factor. d. 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. The options are a. x , b. x − 1 , c. x + 1 , d. No 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 ) So, the factors of x 2 − x are x and ( 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 ) So, the factors of 2 x 2 + x − 1 are ( 2 x − 1 ) and ( x + 1 ) .

Comparing the factors Now, let's compare the factors of both expressions: x 2 − x = x ( x − 1 ) 2 x 2 + x − 1 = ( 2 x − 1 ) ( x + 1 ) The common factor between the two expressions is not immediately obvious. However, let's examine the given options.

Identifying the common factor The factors of x 2 − x are x and x − 1 . The factors of 2 x 2 + x − 1 are ( 2 x − 1 ) and ( x + 1 ) . Comparing the factors, we see that there is no common factor between the two expressions. However, we need to check if any of the options are factors of both expressions. Option a: x . x is a factor of x 2 − x , but not of 2 x 2 + x − 1 .
Option b: x − 1 . x − 1 is a factor of x 2 − x , but not of 2 x 2 + x − 1 .
Option c: x + 1 . x + 1 is a factor of 2 x 2 + x − 1 , but not of x 2 − x .
Therefore, there is no common factor between the two expressions.

Final Answer Therefore, the correct answer is d. No common factor.


Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. In economics, factoring can be used to analyze supply and demand curves. In computer graphics, factoring can be used to optimize rendering algorithms. Understanding common factors helps in simplifying expressions and solving equations efficiently.

Answered by GinnyAnswer | 2025-07-04