Question

Dividing the polynomial P(x) by x-6 yields a quotient by Q(x) and a remainder of 5.

Answer 1

Dividing the given polynomial by (x -6) gives quotient Q(x) and remainder 5 then for Q(-6) = 3 , P(-6) = -31and P(6) =5.

As given in the question,

P(x) be the given polynomial

Dividing P(x) by divisor (x-6) we get,

Quotient = Q(x)

Remainder = 5

Relation between polynomial, divisor, quotient and remainder is given by :

P(x) = Q(x)(x-6) + 5   __(1)

Given Q(-6) = 3

Put x =-6 we get,

P(-6) = Q(-6)(-6-6) +5

⇒ P(-6) = 3(-12) +5

⇒ P(-6) =-36 +5

⇒ P(-6) = -31

Now x =6 in (1),

P(6) = Q(6)(6-6) +5

⇒ P(6) = Q(6)(0) +5

⇒ P(6) = 5

Therefore, dividing the given polynomial by (x -6) gives quotient Q(x) and remainder 5 then for Q(-6) = 3 , P(-6) = -31and P(6) =5.

The complete question is:

Dividing the polynomial P(x) by x - 6 yields a quotient Q(x) and a remainder of 5. If Q(-6) = 3, find P(-6) and P(6).

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