7. Describe the number and type of roots for the polynomial (how many real and complex?)
x^3 + 5x^2 - 4x - 2= 0
sketch graph
at x = -oo it goes to -oo
at x = -1 it is +6
at x = 0 it is-2
at x = +oo it is +oo
SO
it crosses the x axis three times, three real roots
x^3 + 5x^2 - 4x - 2
= (x-1)(x^2+6x+2)
The discriminant of the quadratic is positive, so it has two real roots.
To determine the number and type of roots for the given polynomial (x^3 + 5x^2 - 4x - 2 = 0), we can use the rational root theorem and synthetic division.
Step 1: Rational Root Theorem
The rational root theorem states that if a polynomial has a rational root (a/b), where a is a factor of the constant term and b is a factor of the leading coefficient, then (a/b) will be a root of the polynomial. In this case, the constant term is -2 and the leading coefficient is 1. Therefore, possible rational roots can be found by testing all the factors of -2 divided by the factors of 1.
The factors of -2 are ±1 and ±2, and the factors of 1 are ±1. Therefore, the possible rational roots of the polynomial are ±1, ±2.
Step 2: Synthetic Division
To find the roots, we will use synthetic division with the possible rational roots we obtained from the rational root theorem. Let's start with the first possible rational root, which is 1.
1 | 1 5 -4 -2
_______
1 6 2 0
The result of synthetic division gives us the quotient 1x² + 6x + 2 with a remainder of 0. This means that (x - 1) is a factor of the given polynomial.
Now, we have reduced the polynomial to a quadratic equation: 1x² + 6x + 2 = 0.
Step 3: Quadratic Equation
To solve the quadratic equation, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.
In this case, a = 1, b = 6, and c = 2. Plugging these values into the quadratic formula gives us:
x = (-6 ± √(6² - 4(1)(2))) / (2(1))
x = (-6 ± √(36 - 8)) / 2
x = (-6 ± √28) / 2
x = (-6 ± 2√7) / 2
x = -3 ± √7
Hence, the other two roots of the polynomial are -3 + √7 and -3 - √7.
In summary, the given polynomial (x^3 + 5x^2 - 4x - 2) has one real root (1) and two complex roots (-3 + √7 and -3 - √7).