Question

Which polynomial can be factored using the binomial theorem? 343x^3 – 441x^2 + 567x – 729 343x^3 – 882x^2 + 1,134x – 729 1,296x^4 – 216x^3 + 36x^2 – 6x + 1 1,296x^4 – 864x^3 + 216x^2 – 24x + 1

Answer 1

1,296x^4 – 864x^3 + 216x^2 – 24x + 1

Explanation:

From binomial theorem we know that:

`(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4`

`a^4 = 1,296x^4 = (6x)^4`

a = 6x

`b^4 =1`

b = -1 or b = 1

We have to satisfy:
`4ab^3 = -24x`. Using b = -1

`4(6x)(-1)^3 = 24x(-1) = -24x`

We have to satisfy:
`4a^3b = -864x^3`

`4(6x)^3(-1) = (-4)216x^3 = -864x^3`

We have to satisfy: 6a^2b^2 = 216x^2

`6(6x)^2(-1)^2 = 6(36)x^2 = 216x^2`

Then,
`(6x - 1)^4 = 1,296x^4 - 864x^3 + 216x^2 - 24x + 1`