Question

For the given polynomials, complete the following tasks:
• List the possible rational roots.
• Express the polynomial in factored form.
x^3-x^2+4x-4
3x^4-5x^3-5x^2+5x+2

Answer 1

can someone please show how to do this
for the first one i think the possible rots are 1,2,4
for the second one i think its 1/2, 1, 2
not sure how to do the rest....please help

Answer 2

for the first one i mean negative and positive for 1,2,4

Answer 3

there's a grouping technique which is listed here for cubics

in practise it is worth just guessing and using long division

for the second trickier one

\(3x^4-5x^3-5x^2+5x+2\)

start with x = 1

\(3-5-5+5+2 = 0\) Bingo!!

then with x = -1

\(3+5-5-5+2 = 0\) Bingo!!

\( \dfrac {3x^4-5x^3-5x^2+5x+2} {x^2 -1 } = 3x^2 - 5x -2\)

and apply the quadratic formula unless, unlike me, you can see further roots

\(\dfrac{5 \pm \sqrt{ (-5)^2 - 4(3)(-2) }}{2(3)}\)

\(= \dfrac{5 \pm \sqrt{ 25 + 24 }}{6}\)

gives you \(-\frac{1}{3}, 2\)

is that what you meant?!?!

Answer 4

so -1/3, 2 are my rational roots?

Answer 5

@Nesha97 @IrishBoy123
since they are fractions, then they seem to represent rational numbers (pi is irrational 'cos you can't express it as a fraction), I'd guess that yes is the answer to your q.

Answer 6

the fact that 1/3 is 0.33333333333333333333333 until the end of space-time-ink-paper is slightly embarrasing .... leave it as 1/3 ?

Answer 7

how do i express the polynomial in factored form?

Answer 8

for 4 roots, \(r_1, r_2, r_3, r_4\) its

\(f(x) = (x-r_1)(x-r_2)(x-r_3)(x-r_4)\)

And yes, whole numbers are rational so it's the whole shebang

Answer 9

\(f(x) = (x-1)(x \color{red}{+}1)(x \color{red}{+} \frac{1}{3})(x-2)\)

Answer 10

okay thank you

Answer 11

mp