Factorise cubic equation and solve the inequality?

Question

Please show me the steps to solve this question:
Factorise 2-x+2x^2-3x^3 completely. Hence, solve the inequality 2(1+x^2) < x(1+x)^2

Answer 1

PROBLEM 1:

Let's put the terms in descending order:
-3x^3 + 2x² -x + 2

By inspection, x = 1 will make this zero, so (x - 1) will be a factor. Now use synthetic division:
...................... -3x² -x - 2
x - 1 ) -3x^3 + 2x² -x + 2
......... -3x^3 + 3x²
......... -------------------
....................... -x² - x
....................... -x² + x
...................... ----------------
...................... ...... -2x + 2
...................... ...... -2x + 2

So you have:
(x - 1)(-3x² -x - 2)

PROBLEM 2:
2(1 + x²) < x (1+x)²

Multiply everything out:
2 + 2x² < x (1² + 2x + x²)
2 + 2x < x + 2x² + x^3

Subtract 2x from both sides:
2 < x + x^3

Subtract 2 from both sides:
0 < x^3 + x - 2

Switch the order:
x^3 + x - 2 > 0

Again by inspection we can see that x = 1 is a solution for 0, so factor out (x - 1) by synthetic division:

(x - 1)(x² + x + 2) > 0

x² + x + 2 will always be greater than 0. So that means you need (x - 1) > 0

So ultimately this reduces to
x > 1

Another way to see this is that the graph will cross at x = 1. All values to the left will be negative (below the x-axis) and all values to the right will be positive (above the x-axis). See the attached graph:

Answer 2

Solving Cubic Inequalities

Answer 3

Hi,

2(1+x^2) < x(1+x)^2
2 + 2x² < x(1 + 2x + x²)
2 + 2x² < x + 2x² + x³
0 < x³ + x - 2
0 < (x - 1)(x² + x + 2)
The first factor tells you there is a zero at x = 1, which is the only real zero. Everything to the left of x = 1 is graphed below the x axis while everything to the right is above the x axis. Since you are looking for where x³ + x - 2 > 0, then you want when x > 1.

I hope that helps!! :-)

Answer 4

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RE:
Factorise cubic equation and solve the inequality?
Please show me the steps to solve this question:
Factorise 2-x+2x^2-3x^3 completely. Hence, solve the inequality 2(1+x^2) < x(1+x)^2