Question

Suppose h, k, r, s, t∈Z.Set a=3rt(2s+t) and b=3rs(s+2t).Prove the cubic polynomial

f (x) = (x − h)(x − a − h)(x − b − h) + k
passes through the point (h,k), has integer roots, has local extrema with integer coordinates, and has an inflection

point with integer coordinates.

Answer 1

We have

where   and  .

Since and are integers, so, and are also integers.

Now, putting in the expression of , we obtain

This shows that passes through the point .

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The second statement is false. The equation need not have integer roots.

For example, if we take , then the equation becomes

which has no integer roots.

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Let be a point of local extrema. To prove that is an integer co-ordinate.

Suppose we have shown that is an integer. Then is also an integer because are integers.

So, it is enough to show that is an integer.

Now, since is a point of local extrema, so,

By the product rule of differentiation, we obtain

Using the quadratic formula, we get

after simplification.

Let us now simplify .

We have,

Putting , and , we get

which are integer roots since

So, the points of local extrema have integer co-ordinates.

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Let be an inflection point. To prove that is an integer co-ordinate.

Again, it is enough to show that is an integer.

Since is a point of inflection, we have,

By previous calculations, we have

Putting , , we obtain

which is an integer.

So, the point of inflection have integer co-ordinates.