Question
Consider the polynomial f(x) = 3x3 – 2x2 – 7x – 2.
(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.
(b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
(a) By using the Rational Zero Theorem, list all possible rational zeros of the given polynomial.
(b) Find all of the zeros of the given polynomial. Be sure to show work, explaining how you have found them.
Answers
drwls
Please use a ^ before exponents.
f(x) = 3x^3 -2x^2 -7x -2
The rational zeros are the rational values of x for which f(x) = 0.
Possible (but not necessarily valid)roots are x = +or- 1/3, 2/3, 1, or 2. One of those values that works is x = -1. That means one of the factors of the polynomial is (x+1).
The other factor is
(3x^3 -2x^2 -7x -2)/(x+1)
Synthetic long division will show that factor to be
3x^2 -5x -2 = 0
Use the quadratic formula to solve for the other two roots.
x = (1/6)[5 +/- sqrt(25 +24)]
= (1/6)(12 or -2)
= 2 or -1/3
f(x) = 3x^3 -2x^2 -7x -2
The rational zeros are the rational values of x for which f(x) = 0.
Possible (but not necessarily valid)roots are x = +or- 1/3, 2/3, 1, or 2. One of those values that works is x = -1. That means one of the factors of the polynomial is (x+1).
The other factor is
(3x^3 -2x^2 -7x -2)/(x+1)
Synthetic long division will show that factor to be
3x^2 -5x -2 = 0
Use the quadratic formula to solve for the other two roots.
x = (1/6)[5 +/- sqrt(25 +24)]
= (1/6)(12 or -2)
= 2 or -1/3