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To find the remainder when the polynomial \( 6x^2 + 11x - 3 \) is divided by \( 2x - 1 \), we can apply **synthetic division** or use the **Remainder Theorem**.

### Using the Remainder Theorem:
The Remainder Theorem states that if a polynomial \( f(x) \) is divided by \( (ax - b) \), the remainder is \( f\left(\frac{b}{a}\right) \).

Here, the divisor is \( 2x - 1 \), so we set \( 2x - 1 = 0 \), which gives:
\[
x = \frac{1}{2}
\]

Now, substitute \( x = \frac{1}{2} \) into the polynomial \( f(x) = 6x^2 + 11x - 3 \):
\[
f\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^2 + 11\left(\frac{1}{2}\right) - 3
\]
\[
= 6 \cdot \frac{1}{4} + 11 \cdot \frac{1}{2} - 3
\]
\[
= \frac{6}{4} + \frac{11}{2} - 3
\]
\[
= \frac{3}{2} + \frac{11}{2} - 3
\]
\[
= \frac{14}{2} - 3
\]
\[
= 7 - 3 = 4
\]

Thus, the remainder is **4**.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How do you perform synthetic division with polynomials?
2. What is the Remainder Theorem, and how does it relate to polynomial division?
3. How does long division of polynomials differ from synthetic division?
4. What happens when dividing by a polynomial of higher degree?
5. Can you find the quotient in this polynomial division?

**Tip:** The Remainder Theorem is a quick way to find remainders when dividing polynomials by linear divisors.

What is the remainder when the polynomial 6x^2 + 11x - 3 is divided by 2x - 1?

Learn how to find the remainder when the polynomial 6x^2 + 11x - 3 is divided by 2x - 1 using the Remainder Theorem. This step-by-step solution is suitable for students in Grades 9-12, covering key concepts in algebra and polynomial division.

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