Determine The Remainder When $15x^2 + 19x + 10$ Is Divided By $5x + 3$.A. -6 B. There Is No Remainder. C. 4 D. 16
Introduction
In algebra, polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. The remainder is a polynomial of lesser degree than the divisor. In this article, we will determine the remainder when the polynomial $15x^2 + 19x + 10$ is divided by $5x + 3$.
Polynomial Division
Polynomial division is a long division process similar to the long division of numbers. The process involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.
Step 1: Divide the Highest Degree Term
To begin the division process, we divide the highest degree term of the dividend, which is $15x^2$, by the highest degree term of the divisor, which is $5x$. This gives us $\frac{15x^2}{5x} = 3x$.
Step 2: Multiply the Divisor by the Result
Next, we multiply the entire divisor, $5x + 3$, by the result from Step 1, which is $3x$. This gives us $(5x + 3)(3x) = 15x^2 + 9x$.
Step 3: Subtract the Result from the Dividend
We then subtract the result from Step 2, which is $15x^2 + 9x$, from the dividend, $15x^2 + 19x + 10$. This gives us $(15x^2 + 19x + 10) - (15x^2 + 9x) = 10x + 10$.
Step 4: Repeat the Process
We repeat the process by dividing the highest degree term of the new dividend, which is $10x$, by the highest degree term of the divisor, which is $5x$. This gives us $\frac{10x}{5x} = 2$.
Step 5: Multiply the Divisor by the Result
Next, we multiply the entire divisor, $5x + 3$, by the result from Step 4, which is $2$. This gives us $(5x + 3)(2) = 10x + 6$.
Step 6: Subtract the Result from the Dividend
We then subtract the result from Step 5, which is $10x + 6$, from the new dividend, $10x + 10$. This gives us $(10x + 10) - (10x + 6) = 4$.
Conclusion
After repeating the process, we find that the remainder is $4$. Therefore, the correct answer is:
Introduction
In our previous article, we determined the remainder when the polynomial $15x^2 + 19x + 10$ is divided by $5x + 3$. In this article, we will answer some frequently asked questions related to polynomial division and remainders.
Q: What is polynomial division?
A: Polynomial division is a process of dividing one polynomial by another to obtain a quotient and a remainder. The remainder is a polynomial of lesser degree than the divisor.
Q: How do I perform polynomial division?
A: To perform polynomial division, you need to divide the highest degree term of the dividend by the highest degree term of the divisor, and then multiply the entire divisor by the result and subtract it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.
Q: What is the remainder in polynomial division?
A: The remainder in polynomial division is a polynomial of lesser degree than the divisor. It is the amount left over after the division process is completed.
Q: How do I determine the remainder when a polynomial is divided by another polynomial?
A: To determine the remainder when a polynomial is divided by another polynomial, you need to perform the polynomial division process. This involves dividing the highest degree term of the dividend by the highest degree term of the divisor, and then multiplying the entire divisor by the result and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.
Q: What is the difference between a quotient and a remainder in polynomial division?
A: The quotient in polynomial division is the result of the division process, while the remainder is the amount left over after the division process is completed. The quotient is a polynomial of the same degree as the divisor, while the remainder is a polynomial of lesser degree than the divisor.
Q: Can a polynomial have a remainder of zero?
A: Yes, a polynomial can have a remainder of zero. This occurs when the dividend is exactly divisible by the divisor, and there is no amount left over.
Q: How do I check if a polynomial has a remainder of zero?
A: To check if a polynomial has a remainder of zero, you need to perform the polynomial division process and check if the degree of the remainder is less than the degree of the divisor. If the degree of the remainder is less than the degree of the divisor, then the polynomial has a remainder of zero.
Q: What is the significance of the remainder in polynomial division?
A: The remainder in polynomial division is significant because it can be used to determine the properties of the polynomial, such as its roots and factors. The remainder can also be used to determine the behavior of the polynomial, such as its maximum and minimum values.
Conclusion
In this article, we have answered some frequently asked questions related to polynomial division and remainders. We have discussed the process of polynomial division, the remainder, and the significance of the remainder in polynomial division. We hope that this article has been helpful in understanding the concept of polynomial division and remainders.