Noticing the Pattern of three distinct two variable expressions summing up to zero is the first step
A problem that looks difficult but not awkward, with some symmetry in the expressions, you should be able to detect the key patterns and discover the methods to use the patterns as well as general strategies to solve it quickly in a few steps.
In the problem we have chosen this time, the first thing that catches our attention is the three distinct two variable expressions, $(a-b)$, $(b-c)$ and $(c-a)$ behaving like three variables to form the target expression. Additional bonus is, the sum of the three expressions is zero. This second one complies with three variable zero sum principle. Together, this is a valuable pattern with simple and precise outcome that we know. But the target expression does not directly offer the opportunity to use the simple result associated with the powerful pattern.
Naturally it will be hidden. And that's where the interest in the problem lies.
Watch the video below.
Let us take up process of solving the difficult algebra problem to show you the reasoning of problem solving followed in mind that takes only a short time.
Chosen Problem.
The value of the expression, $\displaystyle\frac{(a-b)^2}{(b-c)(c-a)}+\displaystyle\frac{(b-c)^2}{(a-b)(c-a)}+\displaystyle\frac{(c-a)^2}{(a-b)(b-c)}$ is,
- $2$
- $3$
- $0$
- $\displaystyle\frac{1}{3}$
Solution: Problem analysis and Pattern identification
Identification of the valuable pattern of three distinct two variable expressions, $(a-b)$, $(b-c)$ and $(c-a)$, that sums up to zero, $(a-b)+(b-c)+(c-a)=0$ has been immediate, but the target expression directly didn't offer a way to use the simple results of this pattern. To find a way to use the promising pattern we accept as our primary objective.
The pattern in fact is a compound pattern comprising of two elementary patterns, both powerful,
- The three distinct two variable expressions, $(a-b)$, $(b-c)$ and $(c-a)$ each appear unchanged throughout the main expression. We can easily substitute three expressions with three new dummy variables, $p$, $q$, and $r$ with without changing the nature of the mother expression in any way. This is a powerful pattern combined with method of component expression substitution.
- Also the sum of the three distinct two variable expressions is zero. This is an independent pattern with a simple useful outcome. We classify this pattern as rich pattern. This is the second pattern we identify.
While identifying we look at the two combined together, which by general problem solving principles improves the ease of problem solving significantly.
But as there is no direct method of applying the patterns to reach the simple solution, we shift our direction of attention and resort to the often used highly effective general strategy in simplifying an additive expression comprising of two or more algebraic fractions,
To simplify an additive expression of fractions comprising of two or more terms, we must first find a way to equalize the denominators.
It is so universally used that without applying this strategy we can't even add two fractions, say,
$\displaystyle\frac{1}{4}+\displaystyle\frac{1}{6}$.
For numeric fraction addition, the method is straightforward and standard, just transform both the fractions so that the denominator is changed to the LCM in both cases.
But when in Algebra, Surds or Trigonometry problems such a situation arise, we know we have to equalize the denominators, but, in each problem we have to find out how to do it.
At this point we have formed our immediate objective—we would go for equalizing the three denominators. This is our secondary objective. If we meet this objective, we would be able to sum up the numerators and move towards the solution assuredly.
This is the time we noticed the perfectly symmetric and balanced nature of the three fraction terms in more details.
In a balanced symmetric three variable expression, interchanging two variables won't change the expression.
In each of the terms, we notice the absence of a third expression in the denominator that is similar to the numerator except the power. This is the third and more detailed pattern we observe.
Solution: Stage two—Denominator equalization
Multiplying the three denominators and numerators by $(a-b)$, $(b-c)$ and $(c-a)$ respectively we transform the three denominators to the same value of product of these three distinct two variable expressions.
In the process, the numerators are transformed to cubes of the individual two variable expressions. But the situation is much better now because, with same denominators, these cubes can simply be added together in the form of,
$p^3+q^3+r^3$, where $p=a-b$, $q=b-c$, $r=c-a$, and $p+q+r=0$.
The problem now is reduced to evaluating,
$\displaystyle\frac{p^3+q^3+r^3}{pqr}$, where $p+q+r=0$.
This is what we wanted—a situation perfectly suitable for using the simple outcome of this powerful pattern.
Solution: Stage three—applying the outcome of the pattern
We know, if $p+q+r=0$,
$p^3+q^3+r^3=3pqr$.
This is a very frequently used rich pattern, a pattern with associated simple outcome. It is a rich pattern, where
A simple standard method transforms the pattern into a new simple result.
The simple result bearing method is inbuilt into the pattern. We need not repeat it in all its details, and in this case just use the simple result which is the outcome of the application of the method on the pattern. That's why we call such a pattern as a rich pattern.
We will show the mechanism of it. If you know it, you may skip it.
The rich pattern of zero sum of three variables, with simple powerful outcome
$p+q+r=0$,
Or, $p+q=-r$,
Or, $(p+r)^3=-r^3$
Or, $p^3+q^3+3pq(p+q)=p^3+q^3-3pqr=-r^3$,
Or, $p^3+q^3+r^3=3pqr$.
Substituting this value to the numerator cancels out $pqr$ with the denominator, leaving 3 as the simple result,
$\displaystyle\frac{3pqr}{pqr}=3$.
Answer: Option b: $3$.
Key concepts used: Pattern identification -- Three variable zero sum principle -- Compound pattern -- General fraction simplification strategy -- Symmetric balanced expression -- Denominator equalization -- Abstraction -- component expression substitution -- problem transformation for using the pattern -- Rich pattern with inbuilt method and simple outcome.
With the first step of pattern identification, second step of applying the denominator equalization strategy to the specific problem by a specific method, and the third step of applying the result of the rich pattern all are simple steps that you can easily carry out mentally in say, a few tens of seconds, if you know what to do when, and use the pattern and the method to create the situation where you can use the result of the rich pattern.
That's why we say,
Problem solving in a few steps is basically identifying useful patterns and quickly find and apply methods to use the pattern. It is Patterns and methods.
Remark
To simplify representation of the explanation, we have used single variables $p$, $q$ and $r$ for $(a-b)$, $(b-c)$ and $(c-a)$ respectively. This again is a pattern-method pair. The pattern is,
The two variable expressions appear in the mother expression in unchanged form throughout.
These are simple balanced two variable expressions, and we really didn't need to make the substitution. We did it for ease of explanation.
But occasionally, unless you make the required substitutions of more complex expressions appearing in same form throughout the mother expression by single variables, you may not be able to see the potential of the simplified form.
Such substitutions we call as Component expression substitution, which really is the method associated with this pattern.
The list of Difficult algebra problem solving in a few steps quickly is available at, Quick algebra.
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