(7x3+56x2-14x-42)÷(7x-7)
Introduction
In this article, we will delve into the world of algebraic expressions and solve a complex equation involving polynomials. The given expression is (7x3+56x2-14x-42)÷(7x-7). Our goal is to simplify this expression and find its final value. We will use various algebraic techniques, including factoring and division, to solve this problem.
Understanding the Expression
The given expression is a rational expression, which means it is the ratio of two polynomials. The numerator is a cubic polynomial, while the denominator is a linear polynomial. To simplify this expression, we need to factor both the numerator and the denominator.
Factoring the Numerator
Let's start by factoring the numerator, 7x3+56x2-14x-42. We can factor out the greatest common factor (GCF) of the terms, which is 7. This gives us:
7x3+56x2-14x-42 = 7(x3+8x2-2x-6)
Factoring the Denominator
Now, let's factor the denominator, 7x-7. We can factor out the greatest common factor (GCF) of the terms, which is 7. This gives us:
7x-7 = 7(x-1)
Simplifying the Expression
Now that we have factored both the numerator and the denominator, we can simplify the expression by canceling out any common factors. In this case, we can cancel out the factor of 7 in the numerator and the denominator:
(7x3+56x2-14x-42)÷(7x-7) = (7(x3+8x2-2x-6))÷(7(x-1))
Canceling Out Common Factors
We can cancel out the factor of 7 in the numerator and the denominator:
(7(x3+8x2-2x-6))÷(7(x-1)) = (x3+8x2-2x-6)÷(x-1)
Dividing the Polynomials
Now, we need to divide the polynomials in the numerator and the denominator. We can do this by using long division or synthetic division. Let's use long division:
x3+8x2-2x-6 ÷ x-1
Performing Long Division
To perform long division, we need to divide the leading term of the numerator, x^3, by the leading term of the denominator, x. This gives us x^2. We then multiply the denominator, x-1, by x^2 and subtract the result from the numerator:
x3+8x2-2x-6 - (x3+x2) = 7x^2-2x-6
Continuing the Division
We then bring down the next term, -2x, and divide the leading term of the result, 7x^2, by the leading term of the denominator, x. This gives us 7x. We then multiply the denominator, x-1, by 7x and subtract the result from the numerator:
7x^2-2x-6 - (7x^2-7x) = x-6
Final Result
We have now completed the long division. The final result is:
(x3+8x2-2x-6)÷(x-1) = x^2+7x+6
Conclusion
In this article, we have solved the algebraic expression (7x3+56x2-14x-42)÷(7x-7) by factoring the numerator and the denominator, canceling out common factors, and dividing the polynomials using long division. The final result is x^2+7x+6.
Final Answer
The final answer is x^2+7x+6.
Introduction
In our previous article, we solved the algebraic expression (7x3+56x2-14x-42)÷(7x-7) by factoring the numerator and the denominator, canceling out common factors, and dividing the polynomials using long division. In this article, we will answer some frequently asked questions related to this problem.
Q: What is the greatest common factor (GCF) of the terms in the numerator?
A: The greatest common factor (GCF) of the terms in the numerator is 7.
Q: How do we factor the numerator?
A: We can factor the numerator by factoring out the greatest common factor (GCF) of the terms, which is 7. This gives us:
7x3+56x2-14x-42 = 7(x3+8x2-2x-6)
Q: How do we factor the denominator?
A: We can factor the denominator by factoring out the greatest common factor (GCF) of the terms, which is 7. This gives us:
7x-7 = 7(x-1)
Q: What is the result of canceling out common factors?
A: When we cancel out common factors, we get:
(7(x3+8x2-2x-6))÷(7(x-1)) = (x3+8x2-2x-6)÷(x-1)
Q: How do we divide the polynomials in the numerator and the denominator?
A: We can divide the polynomials in the numerator and the denominator by using long division or synthetic division. In this case, we used long division.
Q: What is the final result of the long division?
A: The final result of the long division is:
(x3+8x2-2x-6)÷(x-1) = x^2+7x+6
Q: What is the final answer to the problem?
A: The final answer to the problem is x^2+7x+6.
Q: Can we use synthetic division instead of long division?
A: Yes, we can use synthetic division instead of long division. Synthetic division is a faster and more efficient method of dividing polynomials.
Q: What are some common mistakes to avoid when solving this problem?
A: Some common mistakes to avoid when solving this problem include:
- Not factoring out the greatest common factor (GCF) of the terms in the numerator and the denominator.
- Not canceling out common factors.
- Not using the correct method of division (e.g. using long division or synthetic division).
- Not checking the final result for errors.
Conclusion
In this article, we have answered some frequently asked questions related to solving the algebraic expression (7x3+56x2-14x-42)÷(7x-7). We hope that this article has been helpful in clarifying any doubts or confusion related to this problem.
Final Answer
The final answer is x^2+7x+6.