Question

1.Factor using difference of cubes (a-b)(a²+ab+b²)

8x³-27

2.Factor by grouping 2x³+7x²+2x+7

3.List the possible roots using the Rational Root Theorem x= +- p/q where q is the leading coefficient and p is the constant

f(x) = 3x⁵ - 2x² - 7x + 4

Answer 1

$1) 8x^3-27=(2x-3)(4x^2)+6x+9)$

$2) 2x^3+7x^2-7x+4=x(x^2+1)+7(x^2+1)=(2x+7)(x^2+1)$

$3) 3x^5-2x^2-7x+4=3m^5/n^5-2m^2/n^2-7m/n+4=0$

$3m^5=2m^2n^3+7mn^4-4n^5$

$3m^5=n^3(2m^2+7mn-4n^2)$

$3m^5/n^3$ divides  entirely

n can be 1, 3, -1 ,-3

(possible denominators of the result)

$3m^5-2m^2n^3-7mn^4=-4n^5$

$m(-3m^4+2mn^3+7n^4)=4n^5$

$4n^5/m$ divides entirely

m can be 1, 4, -1, -4, 2, -2

(possible numerators of the result)

m/n (or p/q) can be the one of combinations of this numbers:

1, -1, 1/3, -1/3, 4, -4, 4/3, -4/3, 2, -2, 2/3, -2/3

Using the substitution method with each of these possiblle roots,

we can find out that m/n=x=2/3.

It is the root of this polynom at which it vanishes, takes a zero value.