factorise the following expression a3+a2+a+1
Answers
Answer:
Pulling out like terms :
2.1 Pull out like factors :
a4 + a3 - a2 - a =
a • (a3 + a2 - a - 1)
Checking for a perfect cube :
2.2 a3 + a2 - a - 1 is not a perfect cube
Trying to factor by pulling out :
2.3 Factoring: a3 + a2 - a - 1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -a - 1
Group 2: a3 + a2
Pull out from each group separately :
Group 1: (a + 1) • (-1)
Group 2: (a + 1) • (a2)
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Add up the two groups :
(a + 1) • (a2 - 1)
Which is the desired factorization
Trying to factor as a Difference of Squares :
2.4 Factoring: a2 - 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 : a2 is the square of a1
Factorization is : (a + 1) • (a - 1)
Multiplying Exponential Expressions :
2.5 Multiply (a + 1) by (a + 1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (a+1) and the exponents are :
1 , as (a+1) is the same number as (a+1)1
and 1 , as (a+1) is the same number as (a+1)1
The product is therefore, (a+1)(1+1) = (a+1)2
Final result :
a • (a + 1)2 • (a - 1)
Answer:
a3 – a2 + a – 1
= a2 (a – 1) + 1(a – 1)
= (a – 1) (a2 + 2)