Question

A) Find a polynomial, which, when added to the polynomial 5x2–3x–9, is equivalent to: 18

b) Find a polynomial, which, when added to the polynomial 5x2–3x–9, is equivalent to: 0

Answer 1

(a)
`-5x^2+3x+27`

(b)
`-5x^2+3x+9`

Explanation:

(a)

Let the polynomial be Q(x).

Given polynomial P(x) =
`5x^2-3x-9`

As per the given statement: A polynomial(Q(x)) which, when added to the polynomial
`5x^2-3x-9`, is equivalent to 18.

`P(x)+Q(x) = 18`

`5x^2-3x-9 +Q(x) = 18`

⇒
`Q(x) = 18 -(5x^2-3x-9)`

or

`Q(x) = 18 -5x^2+3x+9`

Simplify:

`Q(x) =-5x^2+3x+27`

Therefore, the polynomial is,
`-5x^2+3x+27`

Check:

`P(x)+Q(x)` =
`5x^2-3x-9 +(-5x^2+3x+27)`

=
`5x^2-3x-9 -5x^2 +3x+27`

= 18

(b)

Let the polynomial be Q(x).

Given polynomial P(x) =
`5x^2-3x-9`

As per the given statement: A polynomial(Q(x)) which, when added to the polynomial
`5x^2-3x-9`, is equivalent to 0.

`P(x)+Q(x) = 0`

`P(x) = -Q(x)`

⇒
`Q(x) = -(5x^2-3x-9)`

or

`Q(x) = -5x^2+3x+9`

Therefore, the polynomial is,
`-5x^2+3x+9`

Check:

`P(x)+Q(x)`=
`5x^2-3x-9 +(-5x^2+3x+9)`

=
`5x^2-3x-9-5x^2 +3x+9`

= 0

Answer 2

a)
`-5x^(2)+3x+27`

b)
`-5x^(2)+3x+9`

Explanation:

a) Let the required polynomial be p(x).

We have the relation,
`5x^(2)-3x-9` + p(x) = 18

i.e. p(x) = 18
`-5x^(2)+3x+9`

i.e. p(x) =
`-5x^(2)+3x+27`

b) Let the required polynomial be q(x).

We have the relation,
`5x^(2)-3x-9` + q(x) = 0

i.e. q(x) = 0
`-5x^(2)+3x+9`

i.e. q(x) =
`-5x^(2)+3x+9`