Find the rational roots of x^4+3x^3+3x^2-3x-4=0.
A.0,1
B.1,2
C.1,-1
D.-1,2
To find the rational roots of the given equation, we can use the Rational Root Theorem.
The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation, then p must be a factor of the constant term (in this case, -4), and q must be a factor of the leading coefficient (in this case, 1).
So, let's list all the possible rational roots by taking all the factors of -4 and all the factors of 1:
Factors of -4: ±1, ±2, ±4
Factors of 1: ±1
Now, let's test each of these potential rational roots by substituting them into the equation:
When x = 1:
(1)^4 + 3(1)^3 + 3(1)^2 - 3(1) - 4 = 1 + 3 + 3 - 3 - 4 = 0
Therefore, x = 1 is a root of the equation.
When x = -1:
(-1)^4 + 3(-1)^3 + 3(-1)^2 - 3(-1) - 4 = 1 - 3 + 3 + 3 - 4 = 0
Therefore, x = -1 is also a root of the equation.
Hence, the rational roots of the equation x^4 + 3x^3 + 3x^2 - 3x - 4 = 0 are 1 and -1.
Therefore, the correct answer is C. 1, -1.
To find the rational roots of the given equation, you can use the Rational Root Theorem. According to the theorem, any rational root can be expressed as a quotient of factors of the constant term divided by factors of the leading coefficient.
The constant term is -4, and the leading coefficient is 1. So the possible rational roots are fractions in the form ± p/q, where p is a factor of -4 and q is a factor of 1.
The factors of -4 are ± 1, 2, and 4, and the factors of 1 are ± 1.
Therefore, the possible rational roots are:
±1, ±2, ±4.
To find which of these possible roots are actual roots of the equation, you can use synthetic division or substitute each value into the equation to check if it results in zero.
Let's check each possible root:
For x = 0:
0^4 + 3(0^3) + 3(0^2) - 3(0) - 4 = -4 (not zero)
For x = 1:
1^4 + 3(1^3) + 3(1^2) - 3(1) - 4 = 0 (zero)
So, x = 1 is a rational root of the equation.
For x = -1:
(-1)^4 + 3((-1)^3) + 3((-1)^2) - 3(-1) - 4 = 0 (zero)
So, x = -1 is a rational root of the equation.
For x = 2:
2^4 + 3(2^3) + 3(2^2) - 3(2) - 4 = 0 (zero)
So, x = 2 is a rational root of the equation.
Hence, the rational roots of the equation x^4+3x^3+3x^2-3x-4=0 are 1, -1, and 2. Therefore, the correct answer is D. -1, 2.
let f(x) = x^4+3x^3+3x^2-3x-4
f(1) = 1 + 3 + 3 - 3 - 4 = 0
f(-1) = 1 - 3 + 3 + 3 - 4 = 0
f(0) = -4 ≠ 0
f(2) = ..... ≠ 0
so what do you think?