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

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03-11-2026
Mathematics

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

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T02:28:55-05:00

Identify the divisor $2x-1$ and find the zero $x = 1/2$.
Apply the Remainder Theorem by evaluating the polynomial $P(x) = 6x^2 + 11x - 3$ at $x = 1/2$.
Perform the calculation $P(1/2) = 6(1/4) + 11(1/2) - 3 = 1.5 + 5.5 - 3 = 4$.
Confirm the result using polynomial long division to ensure accuracy.
The remainder is $\boxed{4}$.

Explanation

Understanding the Remainder Theorem

We are given the polynomial $P(x) = 6x^2 + 11x - 3$ and we wish to find the remainder when it is divided by the linear divisor $D(x) = 2x - 1$. The Remainder Theorem states that the remainder of a polynomial $P(x)$ divided by a linear factor $(ax - b)$ is equal to $P(\frac{b}{a})$. In our case, the divisor is $2x - 1$, so we set $2x - 1 = 0$, which gives $x = \frac{1}{2}$. Therefore, the remainder will be the value of the polynomial evaluated at $x = \frac{1}{2}$.

Evaluating the Polynomial at $x = 1/2$

To find the remainder, we substitute $x = \frac{1}{2}$ into the original polynomial $P(x) = 6x^2 + 11x - 3$.
Substituting the value:
\[P\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^2 + 11\left(\frac{1}{2}\right) - 3\]
Calculating each term individually:

First term: $6 \left(\frac{1}{4}\right) = \frac{6}{4} = 1.5$
Second term: $11 \left(\frac{1}{2}\right) = \frac{11}{2} = 5.5$
Third term: $-3$

Now sum the results:
\[P\left(\frac{1}{2}\right) = 1.5 + 5.5 - 3\]
\[P\left(\frac{1}{2}\right) = 7 - 3 = 4\]
Thus, the remainder is 4.

Verification by Polynomial Long Division

To verify, we can perform polynomial long division of $6x^2 + 11x - 3$ by $2x - 1$.

Divide the leading term: $6x^2 / 2x = 3x$.
Multiply: $3x \times (2x - 1) = 6x^2 - 3x$.
Subtract: $(6x^2 + 11x - 3) - (6x^2 - 3x) = 14x - 3$.
Divide the new leading term: $14x / 2x = 7$.
Multiply: $7 \times (2x - 1) = 14x - 7$.
Subtract: $(14x - 3) - (14x - 7) = -3 + 7 = 4$.

The quotient is $3x + 7$ and the remainder is 4, which confirms our previous result.

Conclusion

By both applying the Remainder Theorem and performing polynomial long division, we have consistently found that the remainder when $6x^2 + 11x - 3$ is divided by $2x - 1$ is 4. Therefore, the final answer is $\boxed{4}$.

Examples

Polynomial division is a fundamental skill in algebra with applications in fields like engineering and computer science. For example, when designing signal filters, engineers often need to divide transfer functions to simplify complex systems. Calculating the remainder allows them to isolate specific components of a signal, ensuring that electronic circuits function correctly and efficiently. Understanding this division helps in simplifying expressions for easier analysis and solving complex equations.