12x^(5)-3x^(4)-6x-2)÷(x+3)
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Introduction
Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, particularly in solving equations and manipulating expressions. In this article, we will focus on dividing the polynomial 12x(5)-3x(4)-6x-2 by x+3.
Understanding the Problem
To begin with, let's analyze the given polynomial expression: 12x(5)-3x(4)-6x-2. This is a fifth-degree polynomial, meaning it has a term with a degree of 5. The divisor is x+3, which is a linear polynomial. Our goal is to divide the given polynomial by the divisor and simplify the result.
Step 1: Divide the Leading Term
The first step in polynomial division is to divide the leading term of the dividend (12x^(5)) by the leading term of the divisor (x). This will give us the first term of the quotient.
12x^(5) ÷ x = 12x^(4)
Step 2: Multiply and Subtract
Next, we multiply the entire divisor (x+3) by the first term of the quotient (12x^(4)). This will give us a new polynomial that we can subtract from the original dividend.
(12x^(4))(x+3) = 12x^(5) + 36x^(4)
Now, we subtract this new polynomial from the original dividend:
(12x(5)-3x(4)-6x-2) - (12x^(5) + 36x^(4)) = -39x^(4) - 6x - 2
Step 3: Repeat the Process
We now repeat the process by dividing the leading term of the new dividend (-39x^(4)) by the leading term of the divisor (x). This will give us the next term of the quotient.
-39x^(4) ÷ x = -39x^(3)
Step 4: Multiply and Subtract Again
Next, we multiply the entire divisor (x+3) by the next term of the quotient (-39x^(3)). This will give us a new polynomial that we can subtract from the new dividend.
(-39x^(3))(x+3) = -39x^(4) - 117x^(3)
Now, we subtract this new polynomial from the new dividend:
(-39x^(4) - 6x - 2) - (-39x^(4) - 117x^(3)) = 111x^(3) - 6x - 2
Step 5: Continue the Process
We now repeat the process by dividing the leading term of the new dividend (111x^(3)) by the leading term of the divisor (x). This will give us the next term of the quotient.
111x^(3) ÷ x = 111x^(2)
Step 6: Multiply and Subtract Again
Next, we multiply the entire divisor (x+3) by the next term of the quotient (111x^(2)). This will give us a new polynomial that we can subtract from the new dividend.
(111x^(2))(x+3) = 111x^(3) + 333x^(2)
Now, we subtract this new polynomial from the new dividend:
(111x^(3) - 6x - 2) - (111x^(3) + 333x^(2)) = -333x^(2) - 6x - 2
Step 7: Continue the Process
We now repeat the process by dividing the leading term of the new dividend (-333x^(2)) by the leading term of the divisor (x). This will give us the next term of the quotient.
-333x^(2) ÷ x = -333x
Step 8: Multiply and Subtract Again
Next, we multiply the entire divisor (x+3) by the next term of the quotient (-333x). This will give us a new polynomial that we can subtract from the new dividend.
(-333x)(x+3) = -333x^(2) - 999x
Now, we subtract this new polynomial from the new dividend:
(-333x^(2) - 6x - 2) - (-333x^(2) - 999x) = 993x - 2
Step 9: Final Step
We now repeat the process by dividing the leading term of the new dividend (993x) by the leading term of the divisor (x). This will give us the next term of the quotient.
993x ÷ x = 993
Step 10: Multiply and Subtract Again
Next, we multiply the entire divisor (x+3) by the next term of the quotient (993). This will give us a new polynomial that we can subtract from the new dividend.
(993)(x+3) = 993x + 2979
Now, we subtract this new polynomial from the new dividend:
(993x - 2) - (993x + 2979) = -2981
Conclusion
After performing the polynomial division, we obtain the following result:
12x(5)-3x(4)-6x-2)÷(x+3) = 12x^(4) - 39x^(3) + 111x^(2) - 333x + 993
This is the final answer to the given problem.
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Frequently Asked Questions
Q: What is polynomial division?
A: Polynomial division is a mathematical operation that involves dividing one polynomial by another. It is a crucial operation in algebra, particularly in solving equations and manipulating expressions.
Q: Why do we need to perform polynomial division?
A: Polynomial division is necessary when we need to simplify complex expressions or solve equations involving polynomials. It helps us to break down the expression into simpler components and make it easier to work with.
Q: What are the steps involved in polynomial division?
A: The steps involved in polynomial division are:
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by the result from step 1.
- Subtract the result from step 2 from the dividend.
- Repeat steps 1-3 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 the result that is left after performing the division. It is the part of the dividend that cannot be divided by the divisor.
Q: Can the remainder be zero?
A: Yes, the remainder can be zero. This occurs when the dividend is exactly divisible by the divisor.
Q: What is the quotient in polynomial division?
A: The quotient in polynomial division is the result of the division. It is the part of the dividend that is divided by the divisor.
Q: Can the quotient be a polynomial of any degree?
A: Yes, the quotient can be a polynomial of any degree. However, the degree of the quotient is always less than the degree of the dividend.
Q: How do we check if the division is correct?
A: To check if the division is correct, we can multiply the quotient by the divisor and add the remainder. If the result is equal to the dividend, then the division is correct.
Q: What are some common applications of polynomial division?
A: Polynomial division has many applications in mathematics, science, and engineering. Some common applications include:
- Solving equations involving polynomials
- Finding the roots of a polynomial
- Simplifying complex expressions
- Factoring polynomials
- Solving systems of equations
Q: Can polynomial division be performed using technology?
A: Yes, polynomial division can be performed using technology. Many calculators and computer algebra systems have built-in functions for polynomial division.
Q: What are some tips for performing polynomial division?
A: Here are some tips for performing polynomial division:
- Make sure to divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by the result from step 1.
- Subtract the result from step 2 from the dividend.
- Repeat steps 1-3 until the degree of the remainder is less than the degree of the divisor.
- Check your work by multiplying the quotient by the divisor and adding the remainder.
Common Mistakes to Avoid
Mistake 1: Not dividing the leading term of the dividend by the leading term of the divisor.
A: Make sure to divide the leading term of the dividend by the leading term of the divisor.
Mistake 2: Not multiplying the entire divisor by the result from step 1.
A: Make sure to multiply the entire divisor by the result from step 1.
Mistake 3: Not subtracting the result from step 2 from the dividend.
A: Make sure to subtract the result from step 2 from the dividend.
Mistake 4: Not repeating steps 1-3 until the degree of the remainder is less than the degree of the divisor.
A: Make sure to repeat steps 1-3 until the degree of the remainder is less than the degree of the divisor.
Conclusion
Polynomial division is a fundamental concept in algebra that involves dividing one polynomial by another. It is a crucial operation in mathematics, particularly in solving equations and manipulating expressions. By following the steps involved in polynomial division and avoiding common mistakes, we can simplify complex expressions and solve equations involving polynomials.