Question

How do you factor the expression `56x^3 +43x^2+5x`?

Answer 1

`56x^3+43x^2+5x = x(7x+1)(8x+5)`

Explanation:

First separate out the common factor `x`:

`56x^3+43x^2+5x = x(56x^2+43x+5)`

To factor the remaining quadratic expression, use an AC method.

Find a pair of factors of `AC = 56*5 = 280` with sum `B = 43`.

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To help find the appropriate pair you can proceed as follow:

Find the prime factorisation of `280`:

`280 = 2*2*2*5*7`

Next note that `43` is odd, so it is the sum of an odd and an even number.

As a result, the prime factors must be split between the pair in such a way that all factors of `2` are on one side or the other.

This leaves the following possibilities to check the sum:

`1 + 5*7*2^3 = 1 + 280 = 281`

`5 + 7*2^3 = 5 + 56 = 61`

`7 + 5*2^3 = 7 + 40 = 47`

`5*7 + 2^3 = 35 + 8 = 43`

The last pair `35, 8` works.

Use this pair to split the middle term and factor by grouping:

`56x^2+43x+5`

`=56x^2+35x+8x+5`

`=(56x^2+35x)+(8x+5)`

`=7x(8x+5)+1(8x+5)`

`=(7x+1)(8x+5)`

Putting it all together:

`56x^3+43x^2+5x = x(7x+1)(8x+5)`