Polynomial Division: Finding The Quotient Of Two Polynomials
Polynomial division can sometimes feel like navigating a maze, but don't worry! This comprehensive guide will walk you through the process of dividing polynomials step by step, turning what seems complex into a clear and manageable task. We'll specifically address the question of finding the quotient when dividing (x^5 - 3x^3 - 3x^2 - 10x + 15) by (x^2 - 5). So, let's sharpen our pencils and dive in!
Understanding Polynomial Division
Before we jump into the specifics, let's establish a solid foundation. Polynomial division, at its core, is very similar to the long division you learned in elementary school with numbers. The goal is the same: to break down a larger expression (the dividend) into smaller, more manageable parts using another expression (the divisor). The result we're most interested in is the quotient, which tells us how many times the divisor fits into the dividend. Sometimes, we'll also have a remainder, which is the part of the dividend that's left over after the division.
Think of it this way: if you have 17 apples and want to divide them equally among 5 friends, polynomial division is like figuring out that each friend gets 3 apples (the quotient), and there are 2 apples left over (the remainder). The key difference is that instead of numbers, we're working with expressions containing variables and exponents.
The components of polynomial division are:
- Dividend: The polynomial being divided (in our case,
x^5 - 3x^3 - 3x^2 - 10x + 15). - Divisor: The polynomial we are dividing by (in our case,
x^2 - 5). - Quotient: The result of the division (the polynomial that represents how many times the divisor goes into the dividend).
- Remainder: The polynomial left over after the division (if any).
Now that we have these definitions down, let's move on to the actual process of performing the division.
Step-by-Step Guide to Polynomial Division
The method we'll use is called long polynomial division, which mirrors the long division method used with numbers. Here’s a breakdown of the steps involved, illustrated with our specific problem:
1. Set up the division:
Write the dividend (x^5 - 3x^3 - 3x^2 - 10x + 15) inside the division symbol and the divisor (x^2 - 5) outside. It's crucial to include placeholders for any missing terms in the dividend. Notice that there's no x^4 term in our dividend, so we'll add 0x^4 as a placeholder to maintain proper alignment during the division process. This gives us:
________________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
This setup ensures that like terms (terms with the same variable and exponent) will align vertically, making the subtraction steps easier.
2. Divide the leading terms:
Focus on the leading term of the dividend (x^5) and the leading term of the divisor (x^2). Divide the leading term of the dividend by the leading term of the divisor: x^5 / x^2 = x^3. This result (x^3) is the first term of our quotient. Write it above the division symbol, aligned with the x^3 term in the dividend.
x^3____________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
3. Multiply the quotient term by the divisor:
Multiply the first term of the quotient (x^3) by the entire divisor (x^2 - 5): x^3 * (x^2 - 5) = x^5 - 5x^3. Write this result below the dividend, aligning like terms:
x^3____________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
4. Subtract:
Subtract the result obtained in step 3 (x^5 - 5x^3) from the corresponding terms in the dividend (x^5 + 0x^4 - 3x^3). Remember to distribute the negative sign carefully: (x^5 + 0x^4 - 3x^3) - (x^5 - 5x^3) = 2x^3. Bring down the next term from the dividend (-3x^2) to create the new expression to work with:
x^3____________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2
5. Repeat the process:
Now, repeat steps 2-4 using the new expression (2x^3 - 3x^2). Divide the leading term (2x^3) by the leading term of the divisor (x^2): 2x^3 / x^2 = 2x. This is the next term of our quotient. Write it above the division symbol, aligned with the x terms:
x^3 + 2x________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2
Multiply the new quotient term (2x) by the divisor (x^2 - 5): 2x * (x^2 - 5) = 2x^3 - 10x. Write this result below the current expression, aligning like terms:
x^3 + 2x________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2
2x^3 - 10x
Subtract: (2x^3 - 3x^2 - 10x) - (2x^3 - 10x) = -3x^2 + 15. Bring down the next term from the dividend (+15):
x^3 + 2x________________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2 - 10x + 15
2x^3 - 10x
-------------
-3x^2 + 15
6. Repeat again:
Divide the leading term (-3x^2) by the leading term of the divisor (x^2): -3x^2 / x^2 = -3. This is the next term of our quotient. Write it above the division symbol:
x^3 + 2x - 3____________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2 - 10x + 15
2x^3 - 10x
-------------
-3x^2 + 15
Multiply the new quotient term (-3) by the divisor (x^2 - 5): -3 * (x^2 - 5) = -3x^2 + 15. Write this result below the current expression:
x^3 + 2x - 3____________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2 - 10x + 15
2x^3 - 10x
-------------
-3x^2 + 15
-3x^2 + 15
Subtract: (-3x^2 + 15) - (-3x^2 + 15) = 0. We have reached a remainder of 0.
x^3 + 2x - 3____________
x^2 - 5 | x^5 + 0x^4 - 3x^3 - 3x^2 - 10x + 15
x^5 - 5x^3
-----------
2x^3 - 3x^2 - 10x + 15
2x^3 - 10x
-------------
-3x^2 + 15
-3x^2 + 15
-------------
0
7. The Result:
Since the remainder is 0, the quotient is the polynomial we obtained above the division symbol: x^3 + 2x - 3.
Verification
Always verify your work! To check if our quotient is correct, we can multiply the quotient by the divisor and see if we get the dividend. Let's do that:
(x^3 + 2x - 3) * (x^2 - 5) = x^5 - 5x^3 + 2x^3 - 10x - 3x^2 + 15 = x^5 - 3x^3 - 3x^2 - 10x + 15
This matches our original dividend, so our quotient is indeed correct!
Conclusion
Through the process of long polynomial division, we successfully found the quotient when (x^5 - 3x^3 - 3x^2 - 10x + 15) is divided by (x^2 - 5). The quotient is x^3 + 2x - 3. Remember, the key to mastering polynomial division is practice. Work through various examples, and you'll become more comfortable with the steps involved. And always remember to double-check your work by multiplying the quotient and the divisor to ensure it matches the dividend.
For further learning and practice, you can explore resources like Khan Academy's Polynomial Division section.
