Algebraically solve the following inequality

Question

Using factor theorem and synthetic division, algebraically solve the following inequality. Please include an Interval table.

x³+4>3x²

Answer 1

Substitute.

You can start pos or neg. Use fractions, decimals, pos, neg, it's up to you.

As for the table "x" is the top row

"fx" " bottom.

-∞, ∞ Infinity

Answer 2

Being a cubic equation, we have 3 zeroes, so let's find them.

x³-3x²+4=0

The rational root test says only factors of 4 (1, 2, 4) over 1 (1) can be a rational root.

Descartes Rule of Signs says there are 2 positive roots, so there is one negative (since there are 3)

This is text only, so bear with me. You know how to do synthetic division, so the numbers are all here.

Test 1:

1 | 1 -3 0 4

x 1 -2 -2

1 -2 -2 2

Test 2

2 | 1 -3 0 4

x 2 -2 -4

1 -1 -2 0

(x-2) works leaving (x²-x-2)

(x²-x-2) factors into (x-2)(x+1)

We have all three factors (x-2)(x-2)(x+1), so the zeroes are -1 and 2 with a 'bounce' at 2

What happens to x³+4 when x is -M? It is -M, and 3x² is always positive, so everything to the left of -1 is false

What happens between -1 and 2? Check 1. 1³+4>3(1)² --> 5>3, so yes. Between -1 and 2 is true.

Because of the bounce at 2, the truth doesn't change, but check it anyway. What happens to x³+4 when x=M? It is M, and M³ is bigger than M², so everything to the right of 2 is true.

Finally, what happens at -1 and 2? Those points are where the left and right sides are the same, but this is > rather than ≥, so it is false at those points.

Final answer. x³+4>3x² from (-1,2)U(2,∞) ■