Answer: -1, 1
Step-by-step explanation:
To solve the polynomial equation x^4 + 3x^3 + 3x^2 - 3x - 4 = 0, do a quick test of polynomials using 1 and -1 in place of x.
This gives the sum of coefficients as zero
Testing with 1: 1 + 3 + 3 - 3 - 4 = 0
Testing with -1: 1 + (-3) + 3 - (-3) - 4
= 1 - 3 + 3 + 3 - 4 = 0
This gives that both (x + 1) and (x - 1) are roots.
(x + 1)(x - 1) = x² - 1 {difference of two squares}
Dividing the polynomial x^4 + 3x^3 + 3x^2 - 3x - 4 by (x² - 1) results in x² - 3x + 4 as quotient. Factorizing this result would give complex roots.
Therefore, x² - 1 = 0
x² = 1
Taking square of both sides of the equation gives
x = ± 1
1 and -1 are the rational roots to the polynomial.
