Factorise: (i) x 3 − 2x 2 − x + 2 (ii) x 3 + 3x 2 −9x − 5 (iii) x 3 + 13x 2 + 32x + 20 (iv) 2y 3 + y 2 − 2y − 1
Answers
Answer:
Step-by-step explanation:
x3-2x2-x+2
Final result :
(x + 1) • (x - 1) • (x - 2)
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(((x3) - 2x2) - x) + 2
Step 2 :
Checking for a perfect cube :
2.1 x3-2x2-x+2 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-2x2-x+2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -x+2
Group 2: x3-2x2
Pull out from each group separately :
Group 1: (-x+2) • (1) = (x-2) • (-1)
Group 2: (x-2) • (x2)
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Add up the two groups :
(x-2) • (x2-1)
Which is the desired factorization
Trying to factor as a Difference of Squares :
2.3 Factoring: x2-1
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 1 is the square of 1
Check : x2 is the square of x1
Factorization is : (x + 1) • (x - 1)
Final result :
(x + 1) • (x - 1) • (x - 2)