Question

If a quadratic equation of the form ax^2 + c when divided by x and (x + 1) leaves remainder 2 and 4 respectively, then the value of a^2 + c^2 is

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Answer 1

Answer:

The value of $a^2+c^2$ is 8

Step-by-step explanation:

Concept used:

Remainder theorem:

The remainer when P(x) is divided by (x-a) is P(a)

Let , $P(x)=ax^2+c$

When P(x) is divided by x , the remainder is 2

By remainder theorem

P(0)=2

a(0)+c=2

c=2

when P(x) is divided by (x+1), the remainder is 4

By remainder theorem,

P(-1)=4

$a(-1)^2+c=4$

$a+c=4$

$a+2=4$

$a=2$

Now,

$a^2+c^2$

$=2^2+2^2$

$=4+4$

$=8$

Answer 2

Answer:

Value of a² + c² is 8

Step-by-step explanation:

let p(x) = ax² + c

We have to find: value of a² + c²

We use remainder theorem,

Remainder Theorem states that let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x - a, then the remainder is p(a).

Given,

2 is remainder when p(x) divided by x

According to remainder theorem,

So,

p(0) = 2

a × 0² + c = 2

c = 2

Given,

4 is remainder when p(x) divided by x + 1.

According to remainder theorem,

So,

p(-1) = 4

a × (-1)² + c = 4

a + 2 = 4

a = 4 - 2

a = 2

Thus, Value of  a² + c²

                         = 2² + 2²

                         = 4 + 4

                         = 8.

Therefore, Value of a² + c² is 8.