- Find out when the expression is exactly divisible by

Solution 1:

Let where A and B are to be determined in terms of p, q, r, a and b. We can assume so because we know from the fundamental theorem of algebra that the if the LHS has to be of degree three in x, the remaining factor in RHS has to be linear in x.

So, expanding out the RHS of above, we get:

We are saying that the above is true for all values of x: hence, coefficients of like powers of x on LHS and RHS are same; we equate them and get a system of equations:

Hence, we get and or that

Also, so that which means

but and hence,

So, the required conditions are and .

2) Find the condition that may be a perfect square.

Solution 2:

Let where A and B are to be determined in terms of p and q; finally, we obtain the relationship required between p and q for the above requirement.

which is true for all real values of x;

Hence, so or

Also, and hence, or

Also, so that so , which is the required condition.

3) To prove that is a perfect square if and .

Proof 3:

Let

or

More later,

Nalin Pithwa.

PS: Note in the method of undetermined coefficients, we create an identity expression which is true for all real values of x.

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