When P(x) is divided by (x - 1) and (x + 3), the remainders are 4 and 104 respectively. When P(x) is divided by x² - x + 1 the quotient is x² + x + 3 and the remainder is of the form ax + b. Find the remainder.

The remainder is 3x - 4

Step-by-step explanation:

[Remember]

So,

In this case our dividend is always P(x).

Part 1

When the divisor is , the remainder is , so we can say

In order to get rid of "Quotient" from our equation, we must multiply it by 0, so

When solving for , we get

When ,

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Part 2

When the divisor is , the remainder is , so we can say

In order to get rid of "Quotient" from our equation, we must multiply it by 0, so

When solving for , we get

When ,

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Part 3

When the divisor is , the quotient is , and the remainder is , so we can say

From Part 1, we know that , so we can substitute and into

When we do, we get:

We will call equation 1

From Part 2, we know that , so we can substitute and into

When we do, we get:

We will call equation 2

Now we can create a system of equations using equation 1 and equation 2

By adding both equations' right-hand sides together and both equations' left-hand sides together, we can eliminate and solve for

So equation 1 + equation 2:

Now we can substitute into either one of the equations, however, since equation 1 has less operations to deal with, we will use equation 1.

So substituting into equation 1:

Now that we have both of the values for and , we can substitute them into the expression for the remainder.

So substituting and into :

Therefore, the remainder is .

step-by-step explanation: