Question

factorise 4y^3-2y^2-y+3​

Answer 1

Answer:

Step-by-step explanation:

4.1 Pull out like factors :

4y3 - 2y2 - 4y + 2 =

2 • (2y3 - y2 - 2y + 1)

Checking for a perfect cube :

4.2 2y3 - y2 - 2y + 1 is not a perfect cube

Trying to factor by pulling out :

4.3 Factoring: 2y3 - y2 - 2y + 1

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -2y + 1

Group 2: 2y3 - y2

Pull out from each group separately :

Group 1: (-2y + 1) • (1) = (2y - 1) • (-1)

Group 2: (2y - 1) • (y2)

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Add up the two groups :

(2y - 1) • (y2 - 1)

Which is the desired factorization

Trying to factor as a Difference of Squares :

4.4 Factoring: y2 - 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 : y2 is the square of y1

Factorization is : (y + 1) • (y - 1)

Final result :

2 • (y + 1) • (y - 1) • (2y - 1)

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