When a polynomial 𝑓(𝑥) is divided by (𝑥 +

Question

When a polynomial 𝑓(𝑥) is divided by (𝑥 +

2), the quotient is 𝑥2 + 2𝑥 + 1 and the
remainder is 5. Find 𝑓(𝑥)
A. 𝑓(𝑥) = 𝑥3 + 4𝑥2 + 7𝑥 + 5 .
B. 𝑓(𝑥) = 𝑥3 + 4𝑥2 + 𝑥 + 5
C. 𝑓(𝑥) = 𝑥3 + 4𝑥2 + 𝑥 + 7
D. 𝑓(𝑥) = 𝑥3 + 4𝑥2 + 5𝑥 + 7

Answer 1

We can use the division algorithm to solve this problem. We know that:

𝑓(𝑥) = 𝑞(𝑥)(𝑥 + 2) + 5

where 𝑞(𝑥) is the quotient. We also know that the quotient is 𝑥2 + 2𝑥 + 1.

So, we can substitute this into the equation above:

𝑓(𝑥) = (𝑥2 + 2𝑥 + 1)(𝑥 + 2) + 5

Expanding the right side, we get:

𝑓(𝑥) = 𝑥3 + 4𝑥2 + 5𝑥 + 7

Therefore, the answer is (D) 𝑓(𝑥) = 𝑥3 + 4𝑥2 + 5𝑥 + 7.

Answer 2

To find 𝑓(𝑥), we will use the fact that the polynomial is divided by (𝑥 + 2), and the quotient is 𝑥^2 + 2𝑥 + 1 while the remainder is 5.

We can write the polynomial 𝑓(𝑥) as the product of (𝑥 + 2) and the quotient plus the remainder:
𝑓(𝑥) = (𝑥 + 2)(𝑥^2 + 2𝑥 + 1) + 5

Now, let's simplify this by distributing:
𝑓(𝑥) = 𝑥^3 + 2𝑥^2 + 𝑥 + 2𝑥^2 + 4𝑥 + 2 + 5

Combining like terms, we have:
𝑓(𝑥) = 𝑥^3 + 4𝑥^2 + 5𝑥 + 7

Therefore, the correct answer is:

D. 𝑓(𝑥) = 𝑥^3 + 4𝑥^2 + 5𝑥 + 7