How do you factor 15x3−18x6−6x+9x4?
1 Answer
Explanation:
Given:
15x3−18x6−6x+9x4
Let's arrange into standard form - putting the terms in descending order of degree to get:
−18x6+9x4+15x3−6x
Note that all of the terms are divisible by
−18x6+9x4+15x3−6x=−3x(6x5−3x3−5x2+2)
Focusing on the remaining quintic factor, note that the sum of the coefficients is
6−3−5+2=0
We can deduce that
6x5−3x3−5x2+2=(x−1)(6x4+6x3+3x2−2x−2)
Let:
f(x)=6x4+6x3+3x2−2x−2
By the rational zeros theorem, any rational zeros of
That means that the only possible rational zeros are:
±16,±13,±12,±23,±1,±2
Further note that the pattern of the signs of the coefficients of
Note also that the pattern of the signs of the coefficients of
To cut a long story a little shorter, we find that none of the "possible" rational zeros are zeros of
So it's probably best to stick with the rational factorisation we have found:
15x3−18x6−6x+9x4=−3x(x−1)(6x4+6x3+3x2−2x−2)