2x^4 + 11x^3 + x^2 – 10x - 4, what are possible rational roots?

Question

2x^4 + 11x^3 + x^2 – 10x - 4
-what are possible rational roots?
-Actual rational roots?
-Other real roots?
-Divide by (x-r) where r is a root. Repeat with as many roots as you know. Can you find the other roots?

Can someone explain the method of getting the answer?

Answer 1

1
-1/2
-3 + sqrt(5)
-3 - sqrt(5)

Answer 2

Write F(x) = 2x^4 + 11x³ + x² - 10x - 4. And observe that, by trial, F(1) = 0 and F(- ½ ) = 0.

Therefore it follows, from Remainder Theorem, that F(x) has at least two real, rational factors (x - 1) & (2x + 1). To find others, divide F(x), by long division, by the product of (x - 1) & (2x + 1) i.e. by (2x² - x -1).

Thus we get the remaining factor as (x² + 6x + 4). So that all rational factors of F(x) are (x - 1), (2x + 1) & (x² + 6x + 4).

The roots of the quadratic factor x² + 6x + 4 = 0 are - 3 ± √5 which are not rational.

Hence the only possible rational roots of F(x) are x = 1 and x = - ½.

It should now be easier to answer the remaining parts of the question.

Answer 3

To find the possible rational roots, you divide all facors of the constant (-4) by all factors of the leading coefficient (2). So the possible rational roots are:
1, -1, 2, -2, 4, -4, 1/2, -1/2

To find the actual real roots, you have to test using synthetic division. Whatever gives you zero as a remainder will be a root. 1 and -1/2 work.

Using the quadratic formula, the other real roots are : (-3 - sqrt(5)) and (-3 + sqrt(5)).

Answer 4

2 words: Long Division.

Answer 5

f(1) = 0 thus x = 1 is a factor of f(x)
To find other factors, use synthetic division:-
1|2__11__1__-10__-4
"|___2__13___14__4
"|2__13__14__4___0

f(x) = (x - 1)(2x³ + 13x² + 14x + 4)