Urgent Inequality Help Please

**rlarach** · October 12th 2007, 02:39 PM

Prove that the inequality
3a4 - 4a3b + b4 ≥ 0
holds for all real numbers a and b.

THANK YOU VERY MUCH FOR YOUR HELP!!!

**Soroban** · October 12th 2007, 03:33 PM

Hello, rlarach!

Prove that: . $3a^4 - 4a^3b + b^4 \;\geq \;0$ for all real numbers $a$ and $b$

Let $f \:=\:3a^4-4a^3b + b^4$

I used the Factor Theorem a few times.

If $b = a$ , the polynomial has value 0.
. . Hence, $(a -b)$ is a factor of $f.$

Using long division: . $3a^3 -4a^3b + b^4\;=\;(a - b)(3a^3 - a^2b - ab^2 - b^3)$

And we find that $(a-b)$ is a factor of the cubic.

Long division: . $f \;=\;(a-b)(a-b)(3a^2+2ab+b^2)$

. . Hence, we have: . $f \;=\;(a-b)^2\left(2a^2 + [a+b]^2\right)$

We see that: $(a-b)^2 \:\geq\:0$

. . and that: . $2a^2 + (a+b)^2\:\geq\;0$

Therefore, $f \:\geq\:0$ for all real $a,\,b$

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