Solve the given polynomial equation. Use the Rational Zero Theorem and Descartes's Rule of Signs as an aid in obtaining the first root.
2 x cubed minus 5 x squared minus 5 x minus 1 equals 02x3−5x2−5x−1=0
2x^3 - 5x^2 - 5x - 1 = 0
let f(x) = 2x^3 - 5x^2 - 5x - 1
try f(1) = 2 - 5 - 5 - 1 ≠ 0
try f(-1) = -2 - 5 + 5 - 1 ≠ 0
try f(1/2) = 1/4 - 5/4 - 5/2 - 1≠ 0
try f(-1/2) = -1/4 - 5/4 + 5/2 - 1 = 0 , yeahh!
so (2x + 1) is a factor
and x = -1/2
by long algebraic division, which I assume you know how to do
2x^3 - 5x^2 - 5x - 1 = (2x+1)(x^2 - 3x - 1)
so for the other two roots:
x^2 - 3x - 1 = 0
x = (3 ± √13)/2
the rule of signs says that there is at most one positive root, and 3 negative roots.
rational zeros states that any rational root will be ±1 or ±1/2
To solve the polynomial equation 2x^3 - 5x^2 - 5x - 1 = 0, we will use the Rational Zero Theorem and Descartes's Rule of Signs to find the first root.
First, we need to find the possible rational zeros of the polynomial. According to the Rational Zero Theorem, the possible rational zeros are the factors of the constant term (in this case, -1) divided by the factors of the leading coefficient (in this case, 2).
The factors of -1 are ±1, and the factors of 2 are ±1 and ±2. Therefore, the possible rational zeros are ±1/1, ±1/2, ±1/2, and ±1/1.
Now, let's use Descartes's Rule of Signs to determine the number of positive and negative real roots.
Counting the sign changes in the polynomial:
From +2x^3 to -5x^2, we have one sign change.
From -5x^2 to -5x, we have no sign change.
From -5x to -1, we have one sign change.
Hence, there is one sign change in the polynomial, which means that there is one positive real root.
Next, let's evaluate the polynomial at x = -1 and x = 1 to find the number of negative real roots.
For x = -1:
2(-1)^3 - 5(-1)^2 - 5(-1) - 1 = 2 + 5 + 5 - 1 = 11
For x = 1:
2(1)^3 - 5(1)^2 - 5(1) - 1 = 2 - 5 - 5 - 1 = -9
From these evaluations, we see that there is one sign change from x = -1 to x = 1, indicating that there is one negative real root.
Therefore, we have found that the polynomial equation has one positive real root and one negative real root.
To solve the given polynomial equation 2x^3 - 5x^2 - 5x - 1 = 0, we can use the Rational Zero Theorem and Descartes's Rule of Signs to aid in finding the first root.
1. Rational Zero Theorem:
The Rational Zero Theorem states that if a polynomial has a rational root, it will be of the form p/q, where p is a factor of the constant term (-1 in this case) and q is a factor of the leading coefficient (2 in this case).
The factors of the constant term -1 are ±1, and the factors of the leading coefficient 2 are ±1 and ±2. Therefore, the possible rational roots are:
±1/1, ±1/2.
2. Descartes's Rule of Signs:
Descartes's Rule of Signs helps determine the number of positive and negative roots.
Start with the given equation, 2x^3 - 5x^2 - 5x - 1 = 0. Count the sign changes in the coefficients:
- There are no sign changes from positive to negative. (0 to 0 is not considered a sign change)
- There is 1 sign change from negative to positive (in the second term, -5x^2 to -5x).
Based on this information, Descartes's Rule of Signs tells us that there are either 1 positive root or there are 2 positive roots.
3. Test the possible rational roots:
Using the possible rational roots obtained from the Rational Zero Theorem, substitute each value into the equation to test if they are roots.
By trying values from the possible rational roots, we find that x = 1/2 is a root of the equation.
4. Dividing the polynomial by (x - 1/2):
Using polynomial long division or synthetic division, divide the polynomial by (x - 1/2). This will reduce the equation to a quadratic equation.
After performing the division, we get:
(x - 1/2)(2x^2 - 4x - 2) = 0
Simplifying, we have:
2x^2 - 4x - 2 = 0
5. Solve the quadratic equation:
To solve the quadratic equation 2x^2 - 4x - 2 = 0, we can factor it or use the quadratic formula.
The factored form is:
2(x - 2)(x + 1) = 0
Setting each factor equal to zero, we get:
x - 2 = 0 -> x = 2
x + 1 = 0 -> x = -1
Therefore, the roots of the polynomial equation 2x^3 - 5x^2 - 5x - 1 = 0 are x = 1/2, x = 2, and x = -1.