A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms. These polynomials are multiplied and all like terms are combined. The resulting polynomial in x has a maximum of how many nonzero terms?

Resulting polynomial contains a maximum mn positive terms.

Step-by-step explanation:

Given one polynomial contains m nonzero terms and second polynomial contains n nonzero terms.

To show after multiplication and combining similar terms how many positive terms contan by both polynomial.

Now let,

m=1 = (a), n=2 = (x, y) then after multiplication a (x+y) = ax+ay, 2 positive terms. m=2 = (a, b), n=3 = (x, y, z) then after multiplication (a, b) (x+y+z) = ax+ay+az+bx+by+bz, 6 positive terms. m=3 = (a, b, c), n=4 = (x, y, z, t) then after multiplication (a+b+c) (x+y+z) = ax+ay+az+at+bx+by+bz+bt+cx+cy+cz+ct, 12 positive terms.

So we see that after multiplication of m and n positive terms, there are mn positive terms are there.

To prove this we have to apply mathematical induction. So let the statement is true for m=p and n=q number of positive terms, then mn=pq.

We have to show avobe ststement is hold for m+1, n+1. Considering,

(m+1) (n+1) = mn+m+n+1=pq+p+q+1=p (q+1) + 1 (q+1) = (p+1) (q+1)

Hence above statement is true for m+1 and n+1.

Thus there will be mn nonzero terms after multiplication and combine positive terms.