Find all the rational zeros of the polynomial, and then find the irrational zeros

P(x)= 2x^4+15x^3+31x^2+20x+4

http://www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php

To find the rational zeros of a polynomial, we can use the Rational Zero Theorem. According to the theorem, the rational zeros are always of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

For the polynomial P(x) = 2x^4 + 15x^3 + 31x^2 + 20x + 4, the leading coefficient is 2 and the constant term is 4.

Factors of 2: ±1, ±2
Factors of 4: ±1, ±2, ±4

Now, let's find the possible rational zeros by taking all possible combinations of these factors (p/q):

±1/±1, ±1/±2, ±1/±4, ±2/±1, ±2/±2, ±2/±4

Simplifying these fractions gives us the following possible rational zeros:
±1, ±1/2, ±1/4, ±2, ±2/2, ±2/4

Next, we can use synthetic division or long division to test these possible zeros one by one. By using synthetic division or long division, we find that the rational zeros for the given polynomial are:

x = -1/2, -1, -2

To find the irrational zeros, we can use the Rational Zero Theorem, which states that if p/q is a zero (rational or irrational) of a polynomial, then (qx - p) is a factor of the polynomial.

Since we have found the rational zeros of the polynomial, we can divide P(x) by (x + 1/2)(x + 1)(x + 2) to determine the remaining factors and possible irrational zeros.

Performing the division, we get:

P(x) = (x + 1/2)(x + 1)(x + 2)(2x + 2)

To find the remaining zero, we can set (2x + 2) = 0 and solve for x:

2x + 2 = 0
2x = -2
x = -1

Thus, the irrational zero of the polynomial P(x) = 2x^4 + 15x^3 + 31x^2 + 20x + 4 is x = -1.

In summary, the rational zeros of the polynomial are x = -1/2, -1, and -2, while the irrational zero is x = -1.

To find the rational zeros of a polynomial, you need to look for factors of the constant term (i.e., the term without a variable) divided by factors of the leading coefficient (i.e., the coefficient of the highest power term).

In this case, the constant term is 4, and the leading coefficient is 2. Thus, the possible rational zeros can be obtained by dividing factors of 4 by factors of 2.

Factors of 4: ±1, ±2, ±4
Factors of 2: ±1, ±2

Possible rational zeros: ±1, ±2, ±4

To check if these are actually zeros, you can use synthetic division or long division to substitute each potential zero into the polynomial and test if the result is zero. However, in this case, none of the potential rational zeros will work, which means there are no rational zeros for the given polynomial.

Moving on to finding the irrational zeros, it is not possible to determine them directly without advanced algebraic methods. You may need to use numerical approximation techniques, such as the Newton-Raphson method or a graphing calculator, to find approximations for the irrational zeros.

If you don't have access to such methods, then it is not possible to calculate the exact values of the irrational zeros for this polynomial.