How do I multiply a non-absolute value with an absolute value? For example: |x-1| to be multiplied with x^2? I got an answer of x^2|x-1| but I'm pretty sure it's wrong.
Thank you!
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Note that $|x-1|$ is always positive and so is $x^2$? What will then happen? – Dec 04 '17 at 11:31
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Do I simply distribute x^2 to (x-1)? – shhh Dec 04 '17 at 11:32
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How would I answer this problem though: |x-1| + x^2? I split it up into x-1 = x^2 and x-1 = x^2. Not sure if this is right, though. – shhh Dec 04 '17 at 11:36
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divide your problem into two cases, one for $(x-1)\geq0$ and one for $(x-1)<0$ – T C Molenaar Dec 04 '17 at 11:38
2 Answers
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Just $$x^2|x-1|.$$ Alternate forms are: $$|x^3-x^2| = \sqrt{(x^3-x^2)^2} = x^2 \sqrt{(x-1)^2}.$$
T C Molenaar
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If you want to take off the modulus, follow this procedure:
We know $(x-1)\geq 0$ for $x\geq1$ and less than zero for $x<1$. Then, we will have: $$x^2|x-1| = \begin{cases} x^2(x-1) &\mbox{if }x\geq 1\\x^2(1-x)&\mbox{if }x<1\end{cases}$$
Can you do likewise for the case of addition as well?