x2 + x + 1 from x2 -x+ 1
Answers
Answer: x³ - 1
Solution 1: We are given two expressions: the first is (x² + x + 1) and the second (x - 1). We multiply the two term by term till all the terms in the first expression are exhausted. Start with the x² term from the first expression, multiply it by x of the second expression and put down the product. Next you multiply the same x² term by -1 of the second expression and write the result. Repeat the process for the other two terms ( x and +1) in the first expression. Having completed the multiplication process, simplify and arrive at the final result.
∴ (x² + x + 1) (x - 1)
= [x².x + x² (- 1) + x.x + x(-1) + 1.x + 1(-1)]
= x³ - x² + x² - x + x - 1 ,which after cancellation of equal terms,
= x³ - 1 (Proved)
Solution 2: Here we use the relevant formula which may be quoted verbally as follows: The difference of the two cubes of any two quantities is equal to the product of two expressions, one of which is the difference of the two quantities, and the other the sum of their squares increased by their product.
If the two quantities are x and 1,
Then the difference of the cubes of x and 1 = x³ - 1³ = x³ - 1
One expression = difference of x and 1 = x - 1
Other or second expression
= (sum of squares of x and 1 + product of x and 1)
= x² + 1² + x.1 = x² + 1 + x = x² + x + 1
∴ By the above theorem
x³ - 1 = (x² + x + 1) (x - 1)
Or, (x² + x + 1) (x - 1) = x³ - 1 (Proved)
Pradeep