use the rational roots theorem

Question

Use the rational roots theorem to find all the roots of the polynomial 2x^4 -17x^3 +46x^2 -43x +12 =0.

Answer 1

The rational roots theorem won't tell you exactly what the roots of a polynomial are; rather, it gives you a list of possible rational roots which you can check through with either polynomial division or by plugging them in and seeing if they satisfy the original equation.

The Rational Root Theorem tells us we have possible rational roots at:

± (factors of constant term) / (factors of leading coefficient)

For your problem, the constant term 12 has factors ±1, ±2, ±3, ±4, ±6, ±12

The leading coefficient has factors ±1, ±2

Therefore we have possible rational roots at ±1/2, ±1, ,±3/2, ±2, ±3, ±4, ±6, ±12

That's 16 possible roots that we would need to check. Notice we don't double count, for example, 2 even though it emerges from 4/2 and (-4)/(-2).

Typically it's easiest to combine the guess-and-check strategy with polynomial division as follows:

Let's try x=1 to see if it's a root of the original equation:

2(1)^4 -17(1)^3 +46(1)^2 -43(1) +12

2 - 17 + 46 - 43 + 12 = 0

Now that we know x=1 is a root of our equation, we can divide our original polynomial by (x-1) to simplify it. After that, you'll have to do some more guessing and checking to find the next root and repeat the same process until you find all the roots or have used all possible roots (in which case you would have some irrational roots).

For this particular problem, you'd find roots at x=1/2, x=1, x=3, and x=4.

For more specifics, try visiting mathportal.org and playing with their polynomial calculators and taking advantage of their lessons and practice problems. Another tip: when guessing, try going through the integers (whole numbers) first.

Answer 2

12/1, 12/2, 6/1, 6/2, 3/1, 3/2, 2/1, 2/2, 1/1, 1/2 +/- are the choices try them as roots, if one or two work, that narrows down the factoring of what's left after long or sythetic division

the coefficient of the leading term must be divisible by the denominator q

and the constant term must be divisible by the numerator p.

p/q is one or more of the above fractions

2x^4 -17x^3 + 46x^2 -43x + 12 = 0

12 is divisible by 12, 6, 3, 2 and 1

2 is divisible by 2 and 1

try 1, it's easiest

2-17+46-43 +12 = 0. It works

1 is a root

one factor is x-1

by Descartes rule there are a maximum 4 positive real roots

and zero negative real roots

so ignore any negative p/q

that means either 4 real roots or 2 real roots and 2 imaginary roots

4th degree equations have 4 roots, imaginary roots come in conjugate pairs

find one more root and you can use the quadratic equation or factor to find the 3rd and 4th roots

divide that factor into the 4th degree polynomial and try the same approach again

the quotient is a 3rd degree polynomial

2x^3 -15x^2 +31x -12

x=3 is another root

2(27) -15(9)+31(3)-12 = 0

divide the cubic polynomial by x-3

to get a quadratic

2x^2 -9x +4

use the quadratic formula if you can't see the factors

(2x-1)(x-4) =0

the last two positive real roots

are 4 and 1/2

the four roots are 1/2, 1, 3, and 4, all positive real roots

if you graphed the 4th degree polynomial you'd get a W shape where the x axis crosses the W in 4 places. those 4 places are the x intercepts which are the roots, the points (1/2, 0), (1,0), (3,0) and (4,0), solutions to the 4th degree equation, the polynomial set equal to zero.

Answer 3

I've written out the solution here: https://imgur.com/a/x78Jsmw

Remember roots are the same as zeroes or x-intercepts.

The big idea behind the rational roots theorem is that dividing all possible factors of constant term on the right by all possible factors of the leading coefficient on the left gives you a list of the potential roots. If a number is a root of the polynomial, dividing by that root will give a remainder of zero. So one way to check if each potential root in our list actually is a root is to use synthetic division/substitution.

If you have any questions, feel free to ask!