Question

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?

Answer 1

Answer:

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.