Question

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

What polynomial must be subtracted from it to obtain 29x 2 - 7 ?
What polynomial must be added to it to obtain a first degree polynomial?

Answer 1

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

Answer 2

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.`