I’m not sure how to solve the following inequality.
x3 - 5x2 + 2x > -8
Can you please explain how to do it. Thanks in advance.
I’d first bring the 8 to the other side.
x^3 - 5x^2 + 2x + 8 > 0
Notice that 2 is a root of the LHS, so you can factor x-2 out:
(x-2)(x^2 - 3x - 4) > 0
x^2 - 3x - 4 factors to (x-4)(x+1) so we have
(x-4)(x-2)(x+1) > 0
LHS is positive if and only if one of the factors is positive and the others are negative, of all three factors are positive. Can you determine the set of x values that work?
how did you know 2 was a root of the lHS?
@MITer94 sorry I understand you have to sub in numbers to see if it will equal zero. Thanks for all your help. I really appreciate it.
@zxcvbnm1216 no prob.
More specifically, by the rational root theorem, you know that the only possible rational roots are +/- 1, 2, 4, 8.
@MITer94 How would I set up the inequality?
@zxcvbnm1216 Hint: If x = 4, 2, or -1, then LHS = 0, so 4, 2, -1 are not solutions.
What if x > 4, or 2 < x < 4, or -1 < x < 2, etc.? I want you to figure it out since it wouldn’t benefit me (or you) to explain everything. Try plugging in numbers such as x = 5 and see if the LHS is positive or negative.
@MITer94 Ok I got it now thanks. I also do not want you to do it as I’m gonna need to know how to do it next time I see this type of question
I noticed another problem in the Redesigned SAT official practice material that requires the rational root theorem.
Test 4 Section 3 (no calculator) Question 18
x^3-5x^2+2x-10=0
For what real value of x is the equation above true?
By the rational root theorem, the possible roots are +/- 1,2,5,10. Since this is a grid-in, you need a positive answer.
You can see that 5 will work because 5^3=5(5^2) and 2(5)=10.
Not many students I know study the rational root theorem, even when they do pre-calc. I think they learn how to solve cubics by graphing.
@Plotinus not quite true. You can factor to (x-5)(x^2 + 2) = 0 and so the only real solution is x = 5 (other solutions are isqrt(2) and -isqrt(2)).
Are you saying that you CAN answer the question on the non-calc section WITHOUT the rational root theorem? The question posed on the test is to find one real root. You would need to factor etc. only if you want to find all three roots. To find one real root rapidly without a calculator, you need the rational root theorem, no?
My point was that I don’t know ANY students who know how to use the rational root theorem to solve polynomial equations. There may be some who studied the rational root theorem in pre-calc, but they never do much with it because they solve higher-order polynomials by graphing. For Math Level 2, they solve higher polynomials with their algebra calculators.
It looks like students are going to have to brush up on their polynomial function theorems for the new test.
@Plotinus Yes - see my previous solution.
Depends. In general, most polynomials don’t factor nicely or have rational roots. If that is the case, then RRT won’t be of any help, and I would probably graph the equation to find the real roots (note that odd-degree polynomials with real coefficients must have a real root).
For other polynomials, such as the one you posted, it might be easier to simply try factoring it or using other tricks rather than using RRT to guess solutions. Another example is the equation x^4 - 5x^2 + 6 = 0, which factors easily and is solvable without a calculator, but RRT is not of any help here.
For others such as the OP’s question, I didn’t see an obvious way of factoring so I guessed at solutions. Because the RRT gives us a list of all possible rational solutions, you don’t need to guess numbers such as x = 3 or x = 5. Of course, one could easily approximate the answer with a graphing calculator.
One could also (God forbid) use the cubic formula. Idk why one would want to do so on the SAT.
Perhaps - the new SAT seems to have slightly more advanced algebra.
“Depends. In general, most polynomials don’t factor nicely or have rational roots. If that is the case, then RRT won’t be of any help, and I would probably graph the equation to find the real roots (note that odd-degree polynomials with real coefficients must have a real root).”
I was only talking about how a student should answer THIS question UNDER EXAM CONDITIONS, not how to solve other types of polynomial equations, or how to solve polynomial equations in general. x^4 - 5x^2 + 6 = 0 is much easier to factor. Since all the exponents are even, we can rewrite it y^2-5y+6=0.
This is a NO-CALCULATOR problem. The student has about 1 minute to answer the question. You think it is possible to graph the function manually to find the real root in 1 minute?
It looks to me like the rational root theorem is a better choice here. a=1 and 10 does not have many factors.
Factoring is also a good choice if the student is accustomed to factoring equations of this kind. I don’t think even my 700-level students would know how to factor equations like this, including the ones who have completed pre-calc.
As far as I know, the kids are not learning in school how to solve equations like this one without a calculator in one minute.
The algebra on the new test is not just more advanced; some of it is different from what the kids are learning in school.
@Plotinus I wouldn’t want to manually plot y = x^3 - 5x^2 + 2x - 10 if I had just 1 minute. Quite honestly, I would try searching for an optimal solution before actually doing all of the tedious calculations, and I usually encourage my students to do the same (whenever I am tutoring).
Many current SAT math problems can be solved multiple ways - some solutions are much faster than others. I’ve seen lots of difficult level 5 SAT math problems that seem like they take forever, but can be solved very quickly using some creativity and a little bit of know-how.
I’m quite surprised by this - virtually every algebra II/pre-calculus textbook I’ve seen has a section and/or lots of exercises on factoring polynomials.
However I will admit that solving x^3 - 5x^2 + 2x - 10 = 0 (by factoring) is a little more complex than if I simply gave the standard textbook exercise “Factor x^3 - 5x^2 + 2x - 10” as you have that extra step there. But anyone who has taken algebra II should at least have the required knowledge to solve the problem without a calculator, even if it takes a few steps.
“I’m quite surprised by this - virtually every algebra II/pre-calculus textbook I’ve seen has a section and/or lots of exercises on factoring polynomials.”
I have in front of me the Barron’s Let’s Review Algebra 2/Trigonometry, which covers all skills tested by the New York State Regents exam in Algebra 2 and Trig.
There is a section on factoring polynomials, but there is no explanation of how to factor cubics and not one problem about factoring polynomials of the kind in this question.
“But anyone who has taken algebra II should at least have the required knowledge to solve the problem without a calculator, even if it takes a few steps.”
I agree they SHOULD, but they DON’T. I can’t speak about students in general, just my unscientific sample.
Hmm interesting. However I was kind of referring to actual algebra and pre-calc textbooks, not really test prep books. My sample size is probably less than 5.
However, going back to the OP’s question, what type of question would you consider it as? One could claim it is a problem about factoring cubics since it factors nicely. Or perhaps a question involving RRT, or a calculator problem (this is why I was saying many questions can be solved multiple ways).
It is a test prep book for the Regents exam written by New York State, so the book is aligned to the school curriculum.
You are of course right there are many ways to solve this problem.
- If this were a no-calculator question, then the easiest way would be to use RRT and then factor. For factored polynomial and also rational function inequalities. I set up a grid over a number line, drawing a vertical line at each root and a horizontal line for each factor, Then I put in + or - in each space to indicate whether the factor is + or - in each number line zone. Counting vertically whether the -'s are even or odd tells you whether the expression is + or - in that zone.
The number-line grid method for poly/rational inequalities is used in European high schools. They do a lot of no-calculator poly/rational inequalities, much harder than this one. As far as I can tell, the students in US schools are not doing these.
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If this were a non-algebra calculator question (ACT, IB, etc.), I would graph the the function with the calculator.
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If this were a Math Level 2 question, I would use solve on my algebra calculator.