Algebra Division Problems And Solutions
Hey guys! Let's dive into some algebra division problems. This is a crucial topic in algebra, and understanding it thoroughly will set you up for success in more advanced math. We'll break down each problem step by step, so you can see exactly how to tackle these types of questions. Let’s jump right in!
Understanding Algebraic Division
Before we get to the problems, let's quickly recap what algebraic division is all about. In essence, it’s the process of dividing one algebraic expression by another. This involves simplifying fractions with variables, using polynomial long division, and sometimes even synthetic division. The goal is to find the quotient and remainder, just like in regular numerical division.
Key Concepts to Remember:
- Polynomials: Algebraic expressions with variables and coefficients.
- Quotient: The result of the division.
- Remainder: The part left over after division.
- Divisor: The expression we are dividing by.
- Dividend: The expression being divided.
Knowing these terms will help you follow along as we work through the examples.
Problem 6: (m + m - 4m - 4m + m² - 1) ÷ (m³ + m² - 4m - 1)
Okay, let’s start with the first problem. Our aim here is to divide the expression (m + m - 4m - 4m + m² - 1) by (m³ + m² - 4m - 1). The first step is to simplify the dividend.
Simplifying the Dividend
Let's simplify (m + m - 4m - 4m + m² - 1). Combine like terms:
- m + m = 2m
- 2m - 4m = -2m
- -2m - 4m = -6m
So, the simplified dividend is m² - 6m - 1. Now, our problem looks like this:
(m² - 6m - 1) ÷ (m³ + m² - 4m - 1)
Long Division Setup
Now we’re going to perform polynomial long division. Set it up like this:
________
m³+m²-4m-1 | m² - 6m - 1
Performing the Division
This is where it gets a bit tricky, but hang in there! We need to figure out what to multiply (m³ + m² - 4m - 1) by to get the first term of the dividend, which is m². Looking at the leading terms, it’s clear that no single term multiplication will directly give us m². This suggests the quotient is likely to be a fraction or that this expression might not simplify neatly. Let’s proceed cautiously.
Since the degree of the dividend (2) is less than the degree of the divisor (3), this tells us that the quotient will be 0, and the remainder will be the dividend itself.
Final Result
So, the result of the division is:
- Quotient: 0
- Remainder: m² - 6m - 1
Therefore, the expression (m² - 6m - 1) / (m³ + m² - 4m - 1) is already in its simplest form. This sometimes happens, and it's important to recognize when an expression can't be simplified further through division.
Problem 7: (a - a + 10 - 27a + 7a) ÷ (a² + 5 - a)
Next up, we have (a - a + 10 - 27a + 7a) divided by (a² + 5 - a). Again, we start by simplifying the dividend.
Simplifying the Dividend
Let’s simplify (a - a + 10 - 27a + 7a). Combine the like terms:
- a - a = 0
- -27a + 7a = -20a
So, the simplified dividend is -20a + 10. Now, our problem looks like this:
(-20a + 10) ÷ (a² - a + 5)
Long Division Setup
Set up the polynomial long division:
________
a²-a+5 | -20a + 10
Performing the Division
Here, we observe that the degree of the dividend (1) is less than the degree of the divisor (2). This means the division will result in a quotient of 0 and the dividend as the remainder.
Final Result
- Quotient: 0
- Remainder: -20a + 10
Thus, the expression (-20a + 10) / (a² - a + 5) remains in its simplest form. Sometimes, the most crucial step is recognizing when an expression can't be simplified further by division.
Problem 8: (3x - 5xy³ + 3y⁴ - x) ÷ (x² - 2xy + y²)
Now, let's tackle the division of (3x - 5xy³ + 3y⁴ - x) by (x² - 2xy + y²). The first step, as always, is to simplify the dividend.
Simplifying the Dividend
Let's simplify (3x - 5xy³ + 3y⁴ - x). Combine like terms:
- 3x - x = 2x
So, the simplified dividend is -5xy³ + 3y⁴ + 2x. Our problem now looks like this:
(-5xy³ + 3y⁴ + 2x) ÷ (x² - 2xy + y²)
Long Division Setup
Set up the polynomial long division:
_____________
x²-2xy+y² | -5xy³ + 3y⁴ + 2x
Performing the Division
This division is a bit more complex because the terms involve multiple variables and higher degrees. The key here is to organize the terms in descending order of powers for each variable and proceed carefully.
We notice that the degree of the highest term in the dividend (y⁴) is the same as the combined degree of the terms in the divisor (x², xy, y²). This means we can attempt the division. However, the x² term in the divisor doesn't readily divide into any term in the dividend without resulting in fractional powers or negative powers, indicating this might also not simplify neatly.
In such cases, it's often useful to rearrange terms to see if a pattern emerges. However, without a straightforward term to multiply the divisor by to match the leading terms of the dividend, we can infer that the quotient is likely 0, with the entire dividend being the remainder.
Final Result
- Quotient: 0
- Remainder: -5xy³ + 3y⁴ + 2x
Therefore, the expression (-5xy³ + 3y⁴ + 2x) / (x² - 2xy + y²) is in its simplest form.
Problem 9: (2n - 2n⁸ + n - 1) ÷ (n² - 2n + 1)
Now, let’s move on to dividing (2n - 2n⁸ + n - 1) by (n² - 2n + 1). As before, the first step is to simplify the dividend.
Simplifying the Dividend
Let's simplify (2n - 2n⁸ + n - 1). Combine like terms:
- 2n + n = 3n
So, the simplified dividend is -2n⁸ + 3n - 1. Now, our problem looks like this:
(-2n⁸ + 3n - 1) ÷ (n² - 2n + 1)
Long Division Setup
Set up the polynomial long division. It’s essential to include placeholders for missing powers of n to keep the terms aligned correctly:
________________________
n²-2n+1 | -2n⁸ + 0n⁷ + 0n⁶ + 0n⁵ + 0n⁴ + 0n³ + 0n² + 3n - 1
Performing the Division
The highest power of n in the dividend is n⁸, and in the divisor, it’s n². This means we will have a quotient with terms up to n⁶. Let’s start the division process.
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Divide -2n⁸ by n² to get -2n⁶. This is the first term of our quotient.
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Multiply (-2n⁶) by (n² - 2n + 1) to get -2n⁸ + 4n⁷ - 2n⁶.
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Subtract this from the dividend:
-2n⁸ + 0n⁷ + 0n⁶ + 0n⁵ + ... -(-2n⁸ + 4n⁷ - 2n⁶) -------------------------- -4n⁷ + 2n⁶ + 0n⁵ + ... -
Bring down the next term, 0n⁵, to get -4n⁷ + 2n⁶ + 0n⁵.
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Divide -4n⁷ by n² to get -4n⁵. This is the next term of our quotient.
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Multiply (-4n⁵) by (n² - 2n + 1) to get -4n⁷ + 8n⁶ - 4n⁵.
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Subtract this from the current dividend:
-4n⁷ + 2n⁶ + 0n⁵ + ... -(-4n⁷ + 8n⁶ - 4n⁵) ---------------------- -6n⁶ + 4n⁵ + 0n⁴ + ... -
Bring down the next term, 0n⁴, to get -6n⁶ + 4n⁵ + 0n⁴.
-
Divide -6n⁶ by n² to get -6n⁴. This is the next term of our quotient.
We can see a pattern emerging. We continue this process:
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Multiply (-6n⁴) by (n² - 2n + 1) to get -6n⁶ + 12n⁵ - 6n⁴.
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Subtract:
-6n⁶ + 4n⁵ + 0n⁴ + ... -(-6n⁶ + 12n⁵ - 6n⁴) ---------------------- -8n⁵ + 6n⁴ + 0n³ + ... -
Divide -8n⁵ by n² to get -8n³.
-
Multiply (-8n³) by (n² - 2n + 1) to get -8n⁵ + 16n⁴ - 8n³.
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Subtract:
-8n⁵ + 6n⁴ + 0n³ + ... -(-8n⁵ + 16n⁴ - 8n³) ---------------------- -10n⁴ + 8n³ + 0n² + ... -
Divide -10n⁴ by n² to get -10n².
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Multiply (-10n²) by (n² - 2n + 1) to get -10n⁴ + 20n³ - 10n².
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Subtract:
-10n⁴ + 8n³ + 0n² + ... -(-10n⁴ + 20n³ - 10n²) ------------------------ -12n³ + 10n² + 0n + ... -
Divide -12n³ by n² to get -12n.
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Multiply (-12n) by (n² - 2n + 1) to get -12n³ + 24n² - 12n.
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Subtract:
-12n³ + 10n² + 0n - 1 -(-12n³ + 24n² - 12n) ---------------------- -14n² + 12n - 1 -
Divide -14n² by n² to get -14.
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Multiply (-14) by (n² - 2n + 1) to get -14n² + 28n - 14.
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Subtract:
-14n² + 12n - 1 -(-14n² + 28n - 14) --------------------- -16n + 13
We stop here because the degree of the remainder (-16n + 13) is less than the degree of the divisor (n² - 2n + 1).
Final Result
- Quotient: -2n⁶ - 4n⁵ - 6n⁴ - 8n³ - 10n² - 12n - 14
- Remainder: -16n + 13
So, the result of the division is:
(-2n⁸ + 3n - 1) ÷ (n² - 2n + 1) = -2n⁶ - 4n⁵ - 6n⁴ - 8n³ - 10n² - 12n - 14 + (-16n + 13)/(n² - 2n + 1)
This problem illustrates how polynomial long division can be a lengthy but systematic process. Each step builds upon the previous one, making it crucial to keep the terms aligned and organized.
Problem 10: (22a⁶⁴ - 5ab² + abᵇ - 40ab⁵) ÷ (a²b - 2ab² - 1068)
This one looks intimidating, guys, but let’s break it down! We're dividing (22a⁶⁴ - 5ab² + abᵇ - 40ab⁵) by (a²b - 2ab² - 1068). The first step is to simplify the dividend, if possible.
Simplifying the Dividend
Looking at the dividend (22a⁶⁴ - 5ab² + abᵇ - 40ab⁵), there aren't any like terms that can be combined directly. The term abᵇ is a bit unusual because the exponent b is a variable. This term makes simplification challenging without knowing the value of b.
Long Division Setup
Now, let's set up the polynomial long division:
________________________________
a²b - 2ab² - 1068 | 22a⁶⁴ - 5ab² + abᵇ - 40ab⁵
Performing the Division
This division is complex due to the high power of a (64) and the presence of the variable exponent b. The initial term 22a⁶⁴ in the dividend is much larger in degree than the terms in the divisor. This suggests that the quotient will have a high-degree term involving a. However, without knowing the value of b, the division of abᵇ is particularly challenging.
Given the complexity, a full step-by-step long division here would be extremely lengthy and potentially not lead to a simplified form without additional information about b. Generally, for such expressions, we look for simplifications or factorizations that might reduce the complexity. In this case, without further simplification possible by factoring or knowing the value of b, attempting a full long division isn't practical.
Key Observation: The presence of abᵇ makes this problem significantly harder to solve directly through polynomial long division. This term's variable exponent means we cannot combine it with other terms or easily determine how it divides with the divisor.
Final Result
Given the complexities and without additional information, we can say that the division (22a⁶⁴ - 5ab² + abᵇ - 40ab⁵) ÷ (a²b - 2ab² - 1068) remains largely in its original form. It might be possible to express the result as a fraction:
(22a⁶⁴ - 5ab² + abᵇ - 40ab⁵) / (a²b - 2ab² - 1068)
However, without simplifying terms or knowing the value of b, further reduction is not feasible.
Problem 11: (16x - 27y - 24x²y²) ÷ (8x⁸ - 9y⁸ + 6xy² - 12x²y)
Alright, let's tackle the division of (16x - 27y - 24x²y²) by (8x⁸ - 9y⁸ + 6xy² - 12x²y). As usual, we'll start by simplifying the dividend, but in this case, there are no like terms to combine, so we move on to the long division setup.
Long Division Setup
We set up the polynomial long division as follows:
__________________________
8x⁸-9y⁸+6xy²-12x²y | -24x²y² + 16x - 27y
Performing the Division
This division looks intimidating because of the high powers of x and y in the divisor. The key thing to notice here is the degree of the dividend compared to the divisor. The highest degree term in the dividend is x²y², which has a degree of 4 (2 + 2). The divisor, however, has terms like 8x⁸ and -9y⁸, which have a degree of 8.
Since the degree of the dividend (4) is less than the degree of the divisor (8), we can immediately conclude that the quotient will be 0, and the remainder will be the dividend itself.
Final Result
- Quotient: 0
- Remainder: -24x²y² + 16x - 27y
Therefore, the expression (-24x²y² + 16x - 27y) / (8x⁸ - 9y⁸ + 6xy² - 12x²y) remains in its simplest form. This type of situation occurs when the divisor is of a significantly higher degree than the dividend, making the division result trivial.
Final Thoughts
So there you have it, guys! We've worked through some challenging algebraic division problems. Remember, the key is to simplify where possible, set up your long division carefully, and take it one step at a time. Don’t get discouraged if a problem seems tough at first. With practice, you'll get the hang of it. Keep practicing, and you'll become an algebra whiz in no time! If you have any questions, drop them in the comments below. Happy dividing!