Express the Division as a Quotient and Remainder

Oct 13, 2025 by windailys.com 52 views

In the following problem, divide using long division. State the quotient, q(x), and the remainder, r(x).

Problem: Divide using Long Division

In this problem, we will be dividing a polynomial by another polynomial using long division. The dividend is $3x^4 - 4x^2 + 6x$ and the divisor is $x - 3$ . Our goal is to express the division as a quotient, $q(x)$ , and a remainder, $r(x)$ .

Step 1: Set up the Long Division

To begin, we need to set up the long division. We will divide the dividend, $3x^4 - 4x^2 + 6x$ , by the divisor, $x - 3$ . We will start by dividing the leading term of the dividend, $3x^4$ , by the leading term of the divisor, $x$ . This will give us the first term of the quotient.

Step 2: Divide the Leading Term

To divide the leading term of the dividend, $3x^4$ , by the leading term of the divisor, $x$ , we will use the following formula:

\frac{3x^4}{x} = 3x^3

So, the first term of the quotient is $3x^3$ .

Step 3: Multiply and Subtract

Next, we will multiply the entire divisor, $x - 3$ , by the first term of the quotient, $3x^3$ . This will give us a polynomial that we can subtract from the dividend.

3x^3(x - 3) = 3x^4 - 9x^3

We will then subtract this polynomial from the dividend, $3x^4 - 4x^2 + 6x$ , to get a new polynomial.

\begin{array}{r} 3x^3 \\ x - 3 \enclose{longdiv}{3x^4 - 4x^2 + 6x} \\ \underline{3x^4 - 9x^3} \\ 5x^3 - 4x^2 + 6x \end{array}

Step 4: Bring Down the Next Term

We will then bring down the next term of the dividend, which is $-4x^2$ . We will then repeat the process of dividing the leading term of the new polynomial, $5x^3$ , by the leading term of the divisor, $x$ .

Step 5: Divide the Leading Term

To divide the leading term of the new polynomial, $5x^3$ , by the leading term of the divisor, $x$ , we will use the following formula:

\frac{5x^3}{x} = 5x^2

So, the next term of the quotient is $5x^2$ .

Step 6: Multiply and Subtract

Next, we will multiply the entire divisor, $x - 3$ , by the next term of the quotient, $5x^2$ . This will give us a polynomial that we can subtract from the new polynomial.

5x^2(x - 3) = 5x^3 - 15x^2

We will then subtract this polynomial from the new polynomial, $5x^3 - 4x^2 + 6x$ , to get a new polynomial.

\begin{array}{r} 3x^3 + 5x^2 \\ x - 3 \enclose{longdiv}{3x^4 - 4x^2 + 6x} \\ \underline{3x^4 - 9x^3} \\ 5x^3 - 4x^2 + 6x \\ \underline{5x^3 - 15x^2} \\ 11x^2 + 6x \end{array}

Step 7: Bring Down the Next Term

We will then bring down the next term of the dividend, which is $6x$ . We will then repeat the process of dividing the leading term of the new polynomial, $11x^2$ , by the leading term of the divisor, $x$ .

Step 8: Divide the Leading Term

To divide the leading term of the new polynomial, $11x^2$ , by the leading term of the divisor, $x$ , we will use the following formula:

\frac{11x^2}{x} = 11x

So, the next term of the quotient is $11x$ .

Step 9: Multiply and Subtract

Next, we will multiply the entire divisor, $x - 3$ , by the next term of the quotient, $11x$ . This will give us a polynomial that we can subtract from the new polynomial.

11x(x - 3) = 11x^2 - 33x

We will then subtract this polynomial from the new polynomial, $11x^2 + 6x$ , to get a new polynomial.

\begin{array}{r} 3x^3 + 5x^2 + 11x \\ x - 3 \enclose{longdiv}{3x^4 - 4x^2 + 6x} \\ \underline{3x^4 - 9x^3} \\ 5x^3 - 4x^2 + 6x \\ \underline{5x^3 - 15x^2} \\ 11x^2 + 6x \\ \underline{11x^2 - 33x} \\ 39x \end{array}

Step 10: Write the Quotient and Remainder

We have now completed the long division. The quotient is $3x^3 + 5x^2 + 11x$ and the remainder is $39x$ .

Express the Division as a Quotient and Remainder

We can express the division as a quotient and remainder as follows:

\frac{3x^4 - 4x^2 + 6x}{x - 3} = 3x^3 + 5x^2 + 11x + \frac{39x}{x - 3}

The quotient is $3x^3 + 5x^2 + 11x$ and the remainder is $\frac{39x}{x - 3}$ .

Conclusion

In this problem, we used long division to divide a polynomial by another polynomial. We expressed the division as a quotient and remainder, and found that the quotient is $3x^3 + 5x^2 + 11x$ and the remainder is $\frac{39x}{x - 3}$ .
Q&A: Long Division of Polynomials

In the previous article, we used long division to divide a polynomial by another polynomial. We expressed the division as a quotient and remainder, and found that the quotient is $3x^3 + 5x^2 + 11x$ and the remainder is $\frac{39x}{x - 3}$ . In this article, we will answer some common questions about long division of polynomials.

Q: What is long division of polynomials?

A: Long division of polynomials is a method of dividing one polynomial by another polynomial. It is similar to long division of numbers, but with polynomials.

Q: Why do we need to use long division of polynomials?

A: We need to use long division of polynomials to simplify complex expressions and to find the quotient and remainder of a division.

Q: How do I know when to stop dividing?

A: You know when to stop dividing when the degree of the remainder is less than the degree of the divisor.

Q: What is the quotient and remainder in long division of polynomials?

A: The quotient is the result of the division, and the remainder is the amount left over after the division.

Q: How do I write the quotient and remainder?

A: You write the quotient and remainder as follows:

\frac{a(x^n) + b(x^{n-1}) + ... + k}{x - m} = q(x) + \frac{r(x)}{x - m}

where $q(x)$ is the quotient, $r(x)$ is the remainder, and $m$ is the divisor.

Q: What is the difference between long division of polynomials and long division of numbers?

A: The main difference between long division of polynomials and long division of numbers is that polynomials can have multiple terms, whereas numbers only have one term.

Q: Can I use long division of polynomials to divide a polynomial by a constant?

A: Yes, you can use long division of polynomials to divide a polynomial by a constant. In this case, the quotient will be the polynomial and the remainder will be zero.

Q: Can I use long division of polynomials to divide a polynomial by a polynomial with a degree greater than the dividend?

A: No, you cannot use long division of polynomials to divide a polynomial by a polynomial with a degree greater than the dividend. In this case, the division is not possible.

Q: How do I check my work when using long division of polynomials?

A: You can check your work by multiplying the quotient and divisor, and then adding the remainder. If the result is equal to the dividend, then your work is correct.

Q: What are some common mistakes to avoid when using long division of polynomials?

A: Some common mistakes to avoid when using long division of polynomials include:

Not bringing down the next term
Not multiplying the divisor by the correct term
Not subtracting the correct term
Not checking your work

Conclusion

In this article, we answered some common questions about long division of polynomials. We discussed the quotient and remainder, and how to write them. We also discussed some common mistakes to avoid when using long division of polynomials. By following these tips and avoiding common mistakes, you can become proficient in using long division of polynomials to simplify complex expressions and to find the quotient and remainder of a division.