What is the remainder when the polynomial $5 x^2+10 x-15$ is divided by $x+5$?

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03-11-2026
Mathematics

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What is the remainder when the polynomial $5 x^2+10 x-15$ is divided by $x+5$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T02:31:17-05:00

Identify the polynomial as $p(x) = 5x^2 + 10x - 15$ and the divisor as $x + 5$.
Apply the Remainder Theorem, which states the remainder is $p(-5)$ for a divisor $x + 5$.
Evaluate the polynomial at $-5$: $5(-5)^2 + 10(-5) - 15$.
Compute the final result: $125 - 50 - 15 = 60$, so the remainder is $\boxed{60}$.

Explanation

Analysis of the Remainder Theorem

The goal is to find the remainder when the polynomial $p(x) = 5x^2 + 10x - 15$ is divided by the linear binomial $d(x) = x + 5$. According to the Remainder Theorem, the remainder of a polynomial $p(x)$ when divided by a linear factor $(x - c)$ is simply the value of the polynomial evaluated at $x = c$.

Identifying the Constant c

To apply the Remainder Theorem, we first express the divisor $x + 5$ in the form $(x - c)$. By rewriting the divisor as $x - (-5)$, we identify that $c = -5$. Therefore, the remainder of the division is equal to the value of the polynomial $p(-5)$.

Calculating the Remainder

We substitute $x = -5$ into the polynomial $p(x) = 5x^2 + 10x - 15$:
\[p(-5) = 5(-5)^2 + 10(-5) - 15\]
First, we calculate the exponent: $(-5)^2 = 25$.
Next, we perform the multiplications:
\[5 \times 25 = 125\]
\[10 \times (-5) = -50\]
Now, combine these values:
\[125 - 50 - 15 = 75 - 15 = 60\]
The value of the polynomial at $-5$ is 60, which represents the remainder.

Final Conclusion

Based on the evaluation of the polynomial at $x = -5$, we have determined that the remainder is 60. Thus, when $5x^2 + 10x - 15$ is divided by $x + 5$, the remainder is $\boxed{60}$.

Examples

The Remainder Theorem is a powerful tool in algebra used to quickly find remainders without performing full polynomial long division. For instance, in computer science and data transmission, verifying if a large data polynomial is divisible by a specific generator polynomial is essential for error detection in cyclic redundancy checks (CRC). By using this theorem, developers can efficiently compute the remainder (or checksum) of the data stream to ensure its integrity without wasting computational cycles on division.