factorise 2x^3-5x^2+x+2 pls answer fast
Answers
Required Answer:-
Given To Factorise:
- 2x³ - 5x² + x + 2
Solution:
This problem can be solved using factor theorem or without using factor theorem.
Factor Theorem: If f(x) is polynomial and α is a real number and f(α) = 0, then f(x - α) is a factor of f(x)
Let,
➡ f(x) = 2x³ - 5x² + x + 2 . . . . . . (i)
Putting x = 2 in (i), we get,
➡ f(2) = 2 × (2)³ - 5 × (2)² + 2 + 2
➡ f(2) = 16 - 20 + 4
➡ f(2) = 20 - 20
➡ f(2) = 0
As f(2) = 0, therefore,
➡ x - 2 is a factor of the polynomial.
Divide the polynomial by (x - 2)
x - 2) 2x³ - 5x² + x + 2 ( 2x² - x - 1
2x² - 4x²
(-) (+)
----------------
-x² + x
-x² + 2x
(+) (-)
------------------
-x + 2
-x + 2
(+) (-)
----------------
0
Therefore,
2x³ - 5x² + x + 2
= (x - 2)(2x² - x - 1)
= (x - 2)(2x² - 2x + x - 1)
= (x - 2)(2x(x - 1) + 1(x - 1))
= (x - 1)(x - 2)(2x + 1)
This can also be solved without using factor theorem. Here is the solution,
2x³ - 5x² + x + 2
= 2x³ - 2x² - 3x² + 3x - 2x + 2
= 2x²(x - 1) - 3x(x - 1) - 2(x - 1)
= (2x² - 3x - 2)(x - 1)
= (2x² - 4x + x - 2)(x - 1)
= (2x(x - 2) + 1(x - 2))(x - 1)
= (2x + 1)(x - 2)(x - 1)
★ Hence, the factorised form of the polynomial is (x - 1)(x - 2)(2x + 1)
Answer:
- Factorised form is (x - 1)(x - 2)(2x + 1)