Question

Given the polynomial 21x 4 + 3y - 6x 2 + 34

Answer 1

Consider the polynomial $21x^4 + 3y - 6x^2 + 34.$ This polynomial has four terms:

• term $21x^4$ of 4th degree;
• term $3y$  of 1st degree;
• term $-6x^2$ of 2nd degree;
• term $34$ of 0 degree or simply the coefficient without variables.

1. What polynomial must be subtracted from it to obtain $29x^2 - 7?$

You must take off first and second terms at all and add termw with $x^2$ and coefficient. Thus, you have to subtract polynomial $21x^4+3y-35x^2+41$.

Check:

$21x^4 + 3y - 6x^2 + 34-(21x^4+3y-35x^2+41)=21x^4+3y-6x^2+34-21x^4-3y+35x^2-41=x^2(-6+35)+(34-41)=29x^2-7.$

2. What polynomial must be added to it to obtain a first degree polynomial?

If you want to obtain a 1st degree polynomial, then you must take off terms of 4th and 2nd degree. So you have to add $-21x^4+6x^2.$

Check:

$21x^4 + 3y - 6x^2 + 34+(-21x^4+6x^2)=21x^4 + 3y - 6x^2 + 34-21x^4+6x^2=3y+34.$

Answer: 1) $21x^4+3y-35x^2+41$ 2) $-21x^4+6x^2.$

Answer 2

34-6x^2, 21x^4/3y  4 is the degree of the polynomail