Question

Dividing the polynomial P(x) by x-1 yields a quotient Q(x) and a remainder of 10 . If Q(-1)=4, find P(-1) and P(1).

Answer 1

`\[P(-1) = 14 \quad \text{and} \quad P(1) = 12.\]` The remainder theorem states that when a polynomial P(x) is divided by (x - c) , the remainder is P(c) .

Step-by-step explanation:

To find P(-1) andP(1), we'll first understand the relationship between the dividend P(x), the divisor (x-1), the quotient Q(x), and the remainder.

Given that (P(x)divided by x-1 results in a quotient Q(x)and a remainder of 10, we can express this as:

`\[P(x) = Q(x) \cdot (x-1) + 10.\]`

Additionally, it's mentioned that (Q(-1) = 4), which means when x is substituted with -1 n the quotient, it equals 4.

Now, let's use this information to find P(-1) and P(1):

1. **Finding P -1:

Substitute -1 into the equation:

`\[P(-1) = Q(-1) \cdot (-1-1) + 10.\]`

Calculate Q(-1) and solve for P(-1).

2. FindingP(1):

Substitute 1 into the equation:

`\[P(1) = Q(1) \cdot (1-1) + 10.\]`

Calculate (Q(1) and solve for(P(1).

Performing these calculations yields (P(-1) = 14) and (P(1) = 12).

This result signifies that when the polynomial P(x) is divided by x-1 the remainder is 10, and whenQ(x) is evaluated at -1, it equals 4. These values satisfy the conditions provided in the problem.