Divide 4x3 - 3x + 1 by 2x - 1

To solve the division of the polynomial $4x^3 - 3x + 1$ by $2x - 1$, we will use the Long Division method for polynomials.

1. Analyze the problem

We are dividing a cubic polynomial by a linear binomial.

• Dividend: $4x^3 + 0x^2 - 3x + 1$ (Note: We insert $0x^2$ as a placeholder to keep the columns aligned).
• Divisor: $2x - 1$

2. Step-by-Step Long Division

Step 1: Divide the first term of the dividend by the first term of the divisor.

• $4x^3 \div 2x = 2x^2$
• Multiply $2x^2$ by $(2x - 1)$: $2x^2(2x - 1) = 4x^3 - 2x^2$
• Subtract this from the dividend: $(4x^3 + 0x^2) - (4x^3 - 2x^2) = 2x^2$

Step 2: Bring down the next term (-3x) and repeat.

• New expression: $2x^2 - 3x$
• Divide the first term: $2x^2 \div 2x = +x$
• Multiply $x$ by $(2x - 1)$: $x(2x - 1) = 2x^2 - x$
• Subtract: $(2x^2 - 3x) - (2x^2 - x) = -3x + x = -2x$

Step 3: Bring down the final term (+1) and repeat.

• New expression: $-2x + 1$
• Divide the first term: $-2x \div 2x = -1$
• Multiply $-1$ by $(2x - 1)$: $-1(2x - 1) = -2x + 1$
• Subtract: $(-2x + 1) - (-2x + 1) = 0$

3. Conclusion

The quotient is $2x^2 + x - 1$.

Comparing this result to your options: A. $2x^2 - x + 1$ B. $2x^2 - x - 1$ C. $2x^2 + x + 1$ D. $2x^2 + x - 1$

The result matches option D.

Exam Tip

When performing polynomial long division, always ensure your dividend is written in descending order of powers (e.g., $x^3, x^2, x^1, \text{constant}$). If a power is missing (like the $x^2$ term in this question), always write it as $0x^2$. This prevents alignment errors that lead to common mistakes!