Use the rational root theorem with polynomial division if needed to find all the zeroes of:
P(x)=2x^4-3x^3+3x^2+5x-3.
P(x) = 2x^4 - 3x^3 + 3x^2 + 5x - 3
To use rational root theorem, we get all the factors of constant divided by the factors of the leading coefficient (or the term with variable of highest degree). In this case, the constant is -3, and the leading coefficient is 2. Thus,
+/- 1,3 / 1,2
= 1, -1, 3, -3, 1/2 , -1/2 , 3/2 , -3/2
To check which of these are the possible roots, substitute each value of x to the function, and if the answer you got is zero, then that value of x is a root.
After trial and error, the values of x that you'll get are -1 and 1/2. Checking:
x = -1:
P(-1) = 2(-1)^4 - 3(-1)^3 + 3(-1)^2 + 5(-1) - 3
P(-1) = 2 + 3 + 3 - 5 - 3
P(-1) = 0
x = 1/2:
P(1/2) = 2(1/2)^4 - 3(1/2)^3 + 3(1/2)^2 + 5(1/2) - 3
P(1/2) = 1/8 - 3/8 + 3/4 + 5/2 - 3
P(1/2) = -2/8 + 3/4 + 10/4 - 12/4
P(1/2) = -1/4 + 13/4 - 12/4
P(1/2) = 0
Since the highest degree of x in the function is 4, there should be 4 roots. Divide the two factors (from the roots you found) by the function. You should get 2(x^2 - 2x + 3).
From here, you can say that the roots of this quadratic equation are imaginary. Use the quadratic formula to get the remaining roots:
Quadratic formula:
x = (-b +/- sqrt(b^2 - 4ac)) / 2a
hope this helps~ `u`
To find the zeroes of a polynomial using the rational root theorem, we need to consider all possible rational roots of the polynomial. The rational root theorem states that if a rational number p/q is a root of the polynomial P(x), then p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the constant term is -3 and the leading coefficient is 2. Therefore, the possible rational roots can be found by taking the factors of -3 (±1, ±3) and dividing them by the factors of 2 (±1, ±2). The possible rational roots are:
±1/2, ±1, ±3/2, ±3
Now let's use polynomial long division to check whether any of these possible rational roots are actually roots of the polynomial P(x).
Starting with the first possibility, let's try x = 1/2:
____________
2x - 1/2 | 2x^4 - 3x^3 + 3x^2 + 5x - 3
Since the term with the highest degree is 2x^4, we divide 2x^4 by 2x, which gives us x^3 as the quotient. Multiplying the divisor 2x - 1/2 by x^3, we get 2x^4 - x^3/2. Subtracting this from the original polynomial, we get
3x^3 + 3x^2 + 5x - 3 - (2x^4 - x^3/2) = -x^3 + 3x^3 + 3x^2 + 5x - 3 + x^3/2 = (2+x^3/2) + 3x^2 + 5x - 3.
Now we repeat the process with the new polynomial 2 + x^3/2 + 3x^2 + 5x - 3, and try dividing by 2x - 1/2. We continue this process, checking each possible rational root, until we find one that yields no remainder.
After performing polynomial long division for each possible root, we find that the rational root x = 1/2 is indeed a zero of the polynomial P(x). This means that (x - 1/2) is a factor of P(x). We can then use synthetic division or further long division to factor out this root, resulting in a reduced polynomial.
Performing the synthetic division, we get:
1/2 | 2 -3 3 5 -3
+1/2 -1/4 1/8 11/16
___________________
2 -5/2 11/4 83/8 61/16
The resulting polynomial after factoring out x = 1/2 is:
P(x) = (2x - 1)(x^3 - 5/2x^2 + 11/4x + 83/8)
Now we can proceed to find the zeroes of the reduced polynomial (x^3 - 5/2x^2 + 11/4x + 83/8) by repeating the process again with the remaining roots.
Note: The remaining cubic polynomial does not have any rational roots among the possible rational roots we found earlier. So, we need to resort to numerical methods such as graphing or using a calculator to find the approximations of the remaining non-rational roots.