The Leading Term of the Dividend

Oct 13, 2025 by windailys.com 93 views

Introduction

Polynomial division is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we will focus on dividing a cubic polynomial by a linear polynomial. We will use the given polynomial $x^3 - 3x^2 - 10x + 24$ and divide it by $x - 2$ . Our goal is to find the quotient and remainder of this division.

Step 1: Fill in the Missing Number

To begin the division process, we need to fill in the missing number in the polynomial long division table. The table is given as:

{ k \begin{array}{|cccc} 1 & -3 & -10 & 24 \\ \hline \end{array} \}

The missing number is represented by the variable $k$ . To find the value of $k$ , we need to determine the first term of the quotient. We can do this by dividing the leading term of the dividend ( $x^3$ ) by the leading term of the divisor ( $x$ ).

The Leading Term of the Dividend

The leading term of the dividend is $x^3$ . This term will be divided by the leading term of the divisor, which is $x$ . Therefore, the first term of the quotient is $x^2$ .

The Leading Term of the Divisor

The leading term of the divisor is $x$ . This term will be multiplied by the first term of the quotient ( $x^2$ ) to obtain the first term of the product.

The First Term of the Product

The first term of the product is obtained by multiplying the first term of the quotient ( $x^2$ ) by the leading term of the divisor ( $x$ ). This gives us $x^3$ .

Filling in the Missing Number

Now that we have determined the first term of the product, we can fill in the missing number in the polynomial long division table. The table now looks like this:

{ \begin{array}{|cccc} x^2 & -3 & -10 & 24 \\ \hline x^3 & -3x^2 & -10x & 24 \\ \hline \end{array} \}

Step 2: Subtract the Product

The next step in the division process is to subtract the product from the dividend. This is done by subtracting the product ( $x^3 - 3x^2 - 10x + 24$ ) from the dividend ( $x^3 - 3x^2 - 10x + 24$ ).

The Result of the Subtraction

The result of the subtraction is:

{ \begin{array}{|cccc} x^2 & -3 & -10 & 24 \\ \hline x^3 & -3x^2 & -10x & 24 \\ \hline 0 & 0 & 0 & 0 \\ \end{array} \}

Step 3: Bring Down the Next Term

Since the result of the subtraction is zero, we can bring down the next term from the dividend. The next term is $-10x$ .

The New Table

The new table now looks like this:

{ \begin{array}{|cccc} x^2 & -3 & -10 & 24 \\ \hline x^3 & -3x^2 & -10x & 24 \\ \hline -10x & 0 & 0 & 0 \\ \end{array} \}

Step 4: Repeat the Process

We can repeat the process of subtracting the product from the dividend. This will give us a new table with the next term of the quotient.

The New Table

The new table now looks like this:

{ \begin{array}{|cccc} x^2 & -3 & -10 & 24 \\ \hline x^3 & -3x^2 & -10x & 24 \\ \hline -10x & 0 & 0 & 0 \\ \hline \end{array} \}

The Final Result

The final result of the division is:

{ \begin{array}{|cccc} x^2 & -3 & -10 & 24 \\ \hline x^3 & -3x^2 & -10x & 24 \\ \hline -10x & 0 & 0 & 0 \\ \hline \end{array} \}

The quotient is $x^2 - 3$ and the remainder is $0$ .

Conclusion

In this article, we have shown how to divide a cubic polynomial by a linear polynomial using polynomial long division. We have filled in the missing number in the polynomial long division table, subtracted the product from the dividend, and brought down the next term from the dividend. The final result of the division is the quotient and remainder.

Final Answer

The final answer is $\boxed{x^2 - 3}$ .

Discussion

Polynomial division is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we have shown how to divide a cubic polynomial by a linear polynomial using polynomial long division. The quotient is $x^2 - 3$ and the remainder is $0$ . This result can be used to solve equations and simplify complex expressions.

References

[1] "Polynomial Long Division" by Math Open Reference
[2] "Cubic Polynomials" by Wolfram MathWorld
[3] "Linear Polynomials" by Math Is Fun

Keywords

Polynomial long division
Cubic polynomials
Linear polynomials
Algebra
Mathematics
Division
Quotient
Remainder
Simplification
Complex expressions
Equations
Dividing Polynomials: A Q&A Guide =====================================

Introduction

In our previous article, we discussed how to divide a cubic polynomial by a linear polynomial using polynomial long division. In this article, we will answer some frequently asked questions about polynomial division.

Q: What is polynomial long division?

A: Polynomial long division is a method of dividing a polynomial by another polynomial. It is similar to long division in arithmetic, but it is used to divide polynomials instead of numbers.

Q: How do I know when to use polynomial long division?

A: You should use polynomial long division when you need to divide a polynomial by another polynomial. This can be useful in solving equations and simplifying complex expressions.

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

A: The quotient is the result of the division, and the remainder is the amount left over after the division. In polynomial long division, the quotient is a polynomial and the remainder is a polynomial or zero.

Q: How do I fill in the missing number in the polynomial long division table?

A: To fill in the missing number in the polynomial long division table, you need to determine the first term of the quotient. This is done by dividing the leading term of the dividend by the leading term of the divisor.

Q: What is the leading term of the dividend and the divisor?

A: The leading term of the dividend is the term with the highest degree, and the leading term of the divisor is the term with the highest degree.

Q: How do I subtract the product from the dividend?

A: To subtract the product from the dividend, you need to multiply the first term of the quotient by the divisor and subtract the result from the dividend.

Q: What is the final result of the division?

A: The final result of the division is the quotient and remainder.

Q: Can I use polynomial long division to divide a polynomial by a polynomial with a degree greater than 1?

A: Yes, you can use polynomial long division to divide a polynomial by a polynomial with a degree greater than 1. However, the process may be more complex and may require multiple steps.

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

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

Not filling in the missing number in the polynomial long division table correctly
Not subtracting the product from the dividend correctly
Not bringing down the next term from the dividend correctly
Not repeating the process correctly

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

A: To check your work when using polynomial long division, you can multiply the quotient by the divisor and add the remainder to the dividend. If the result is the original dividend, then your work is correct.

Q: What are some real-world applications of polynomial long division?

A: Polynomial long division has many real-world applications, including:

Solving equations
Simplifying complex expressions
Finding the roots of a polynomial
Factoring polynomials

Conclusion

In this article, we have answered some frequently asked questions about polynomial long division. We have discussed how to fill in the missing number in the polynomial long division table, subtract the product from the dividend, and bring down the next term from the dividend. We have also discussed how to check your work and some real-world applications of polynomial long division.

Final Answer

The final answer is $\boxed{0}$ .

Discussion

Polynomial long division is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we have discussed how to divide a cubic polynomial by a linear polynomial using polynomial long division. We have also answered some frequently asked questions about polynomial long division.

References

[1] "Polynomial Long Division" by Math Open Reference
[2] "Cubic Polynomials" by Wolfram MathWorld
[3] "Linear Polynomials" by Math Is Fun

Keywords

Polynomial long division
Cubic polynomials
Linear polynomials
Algebra
Mathematics
Division
Quotient
Remainder
Simplification
Complex expressions
Equations