Find the remainder when X3 - 2X2 + 3X - 3 is divided by X2 + 1

To solve this problem, we will use the Polynomial Long Division method. This is the most reliable way to find the remainder when dividing one polynomial by another.

1. Analyze the Question

We are dividing the polynomial $P(x) = x^3 - 2x^2 + 3x - 3$ by the divisor $D(x) = x^2 + 1$. We need to find the remainder.

2. Explain the Concept

When dividing polynomials, we seek a quotient $Q(x)$ and a remainder $R(x)$ such that: $P(x) = D(x) \cdot Q(x) + R(x)$ The degree of the remainder $R(x)$ must be less than the degree of the divisor $D(x)$. Since our divisor is of degree 2 ($x^2$), our remainder will be of the form $ax + b$.

3. Show the Solution

Step 1: Divide the first term of the dividend by the first term of the divisor. Divide $x^3$ by $x^2$: $x^3 \div x^2 = x$ (This is the first term of our quotient).

Step 2: Multiply the divisor by this term and subtract from the dividend. $(x^2 + 1) \cdot x = x^3 + x$ Subtracting this from the original polynomial: $(x^3 - 2x^2 + 3x - 3) - (x^3 + 0x^2 + x)$ $= -2x^2 + 2x - 3$

Step 3: Divide the first term of the new expression by the first term of the divisor. Divide $-2x^2$ by $x^2$: $-2x^2 \div x^2 = -2$ (This is the second term of our quotient).

Step 4: Multiply the divisor by this term and subtract. $(x^2 + 1) \cdot (-2) = -2x^2 - 2$ Subtracting this from our current expression: $(-2x^2 + 2x - 3) - (-2x^2 + 0x - 2)$ $= 2x - 3 + 2$ $= 2x - 1$

Since the degree of $2x - 1$ (which is 1) is less than the degree of $x^2 + 1$ (which is 2), we stop here. The remainder is $2x - 1$.

4. Identify the Answer

Comparing our result to the given options: A. $2x - 1$ B. $x + 3$ C. $2x + 1$ D. $x - 3$

The correct option is A.

5. Exam Tip

When performing polynomial division, always ensure your polynomials are written in descending order of powers (e.g., $x^3, x^2, x^1, x^0$). If a term is missing (like if there was no $x$ term), use a placeholder like $0x$ to keep your columns aligned and avoid silly subtraction errors!