Question

If (x+2) is a factor of p(x)=ax^3+bx^2+x-6 and p(x) when divided by(x-2) leaves the remainder 4. Prove that a=0 and b=2

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Step-by-step explanation:

Given :-

(x+2) is a factor of p(x)=ax^3+bx^2+x-6 and p(x) when divided by(x-2) leaves the remainder 4.

To find :-

Prove that a=0 and b=2

Solution :-

Given Polynomial is ax³+bx²+x-6

Let P(x) = ax³+bx²+x-6

Given factor of P(x) = (x+2)

we know that ,by Factor Theorem

If (x-a) is a factor of P(x) then P(a) = 0

Now,

If (x+2) is a factor of P(x) then P(-2) = 0

=> a(-2)³+b(-2)²+(-2)-6 = 0

=> a(-8)+b(4)-2-6 = 0

=> -8a+4b-8 = 0

=> 4(-2a+b-2) = 0

=> -2a+b-2 = 0/4

=> -2a+b-2= 0

=> b = 2a+2 -----------(1)

and

Given that

P(x) is divided by(x-2) leaves the remainder 4.

By Remainder Theorem

If P(x) is divided by x-a then the remainder is P(a).

Now,

P(x) is divided by(x-2) then the remainder = P(2).

According to the given problem.

P(2) = 4

=> a(2)³+b(2)²+2-6 = 4

=> a(8)+b(4)-4 = 4

=> 8a+4b-4 = 4

=> 8a+4b = 4+4

=> 8a+4b = 8

=> 4(2a+b) = 8

=> 2a+b = 8/4

=>2a+b = 2

=> 2a+2a+2 = 2

=> 4a +2 = 2

=> 4a = 2-2

=> 4a = 0

=> a = 0/4

=> a = 0

On substituting the value of a in (1) then

=> b = 2a+2

=> b = 2(0)+2

=> b = 0+2

=> b = 2

Therefore, a = 0 and b = 2

Hence, Proved.

Answer:-

The value of a = 0 and b = 2

Used formulae:-

Remainder Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if P (x) is divided by (x-a) then the remainder is P(a).

Factor Theorem:-

Let P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if (x-a) is a factor of P (x) then P(a) = 0 vice-versa.