Question

Let f(x) be a polynomial function. If f(x) is divided by x−1,x+1 and x+2, then remainders are 5,3 and 2 respectively. When f(x) is divided by x^3+2x^2−x−2, then remainder is

Answer 1

Answer:

Which of the following option makes the statement below true?

+ secx sec x

cos²x-1-tan²x

=?

Answer 2

Answer:

Correct option is

B

x+4

We know that when a polynomial f(x) is divided by (x−a), the remainder is f(a)

Let f(x) be the polynomial, q(x) be the quotient and r(x) be the remainder.

Degree of remainder is always less than divisor. So, let r(x)=ax

2

+bx+c

Given that f(1)=5; f(−1)=3; f(−2)=2

⇒x

3

+2x

2

−x−2=(x−1)(x+1)(x+2)

⇒f(x)=q(x)×(x−1)(x+1)(x+2)+r(x)

⇒f(1)=q(1)×0+r(1)

⇒5=r(1) ..(1)

⇒f(−1)=q(−1)×0+r(−1)

⇒3=r(−1) ..(2)

⇒f(−2)=q(−2)×0+r(−2)

⇒2=r(−2) ..(3)

Substituting these values from (1),(2),(3) in r(x) we get

⇒r(x)=ax

2

+bx+c

⇒r(1)=a+b+c=5

⇒r(−1)=a−b+c=3

⇒r(−2)=4a−2b+c=2

Solving for a,b,c we get

⇒a=0;b=1;c=4

Therefore, r(x)=x+4

Step-by-step explanation:

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