Question

on factorisinga3+a2+a we GET​

Answer 1

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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)

$\huge{hope \: this \: helps}$