Polynomial Division: P(x) = X^4 + 2x^3 - 18x By D(x) = X - 3
Hey guys! Let's dive into a classic polynomial division problem. We're given two polynomials, P(x) = x^4 + 2x^3 - 18x and D(x) = x - 3, and our mission, should we choose to accept it (and we do!), is to divide P(x) by D(x). But that's not all! We need to express P(x) in a specific form: P(x) = D(x) * Q(x) + R(x). This might look a bit intimidating at first, but trust me, it's just like regular division, only with polynomials! We'll use either synthetic division or long division to crack this nut. So, buckle up, because we're going on a polynomial adventure!
Understanding Polynomial Division
Before we jump into the calculations, let's take a moment to understand what we're actually doing. Polynomial division is a process that allows us to divide one polynomial (the dividend, P(x) in our case) by another polynomial (the divisor, D(x)). The result of this division gives us two things: the quotient, Q(x), which is the result of the division, and the remainder, R(x), which is what's left over after the division. The form P(x) = D(x) * Q(x) + R(x) is simply a way of expressing this relationship. It says that the original polynomial, P(x), can be reconstructed by multiplying the divisor, D(x), by the quotient, Q(x), and then adding the remainder, R(x). This is super important in algebra and calculus, and mastering this now will definitely pay off later. Think of it like this: if you divide 25 by 7, you get a quotient of 3 and a remainder of 4. This means 25 = 7 * 3 + 4. We're doing the same thing, but with polynomials!
Choosing Our Weapon: Synthetic vs. Long Division
Now, let's talk about the methods we can use to perform this division. We have two main options: synthetic division and long division. Both methods will get us to the same answer, but they have their own strengths and weaknesses. Long division is the more general method, and it works for dividing by any polynomial, no matter how complicated. It's also the method that most closely resembles the long division you learned in elementary school with numbers, so it might feel more familiar. Synthetic division, on the other hand, is a shortcut that only works when you're dividing by a linear polynomial of the form (x - c), where c is a constant. In our case, D(x) = x - 3, so we can use synthetic division, and it will be a bit faster and easier. However, since it's good to be versatile, let's walk through both methods so we're prepared for any polynomial division challenge that comes our way. For this specific problem, I think synthetic division will be our best bet because it's quicker and less prone to errors when dealing with a linear divisor like (x - 3). But don't worry, we'll cover long division too, just so you have both tools in your arsenal!
Method 1: Synthetic Division (The Speedy Gonzales Method)
Since our divisor is D(x) = x - 3, we can use the super-efficient synthetic division method. Here's how it works, step by step:
- Identify 'c': In D(x) = x - 3, the value of 'c' is 3. This is the number we'll use in our synthetic division setup.
- Set up the synthetic division: Write down the coefficients of P(x) = x^4 + 2x^3 - 18x. Remember to include a '0' for any missing terms. In this case, we have x^4, x^3, no x^2 term (so we use 0), x, and a constant term (which is also 0). So our coefficients are 1, 2, 0, -18, and 0. Draw a horizontal line and a vertical line to create a little box for our numbers.
3 | 1 2 0 -18 0
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- Bring down the first coefficient: Bring down the first coefficient (which is 1) below the horizontal line.
3 | 1 2 0 -18 0
---------------------
1
- Multiply and add:
- Multiply the number you just brought down (1) by 'c' (3), which gives you 3. Write this under the next coefficient (2).
- Add 2 and 3, which gives you 5. Write this below the line.
3 | 1 2 0 -18 0
---------------------
1 5
- Repeat: Repeat the multiply-and-add process for the remaining coefficients:
- Multiply 5 by 3, which gives you 15. Write this under the next coefficient (0).
- Add 0 and 15, which gives you 15. Write this below the line.
3 | 1 2 0 -18 0
---------------------
1 5 15
* Multiply 15 by 3, which gives you 45. Write this under the next coefficient (-18).
* Add -18 and 45, which gives you 27. Write this below the line.
3 | 1 2 0 -18 0
---------------------
1 5 15 27
* Multiply 27 by 3, which gives you 81. Write this under the last coefficient (0).
* Add 0 and 81, which gives you 81. Write this below the line.
3 | 1 2 0 -18 0
---------------------
1 5 15 27 81
-
Interpret the results: The numbers below the line (except the last one) are the coefficients of the quotient, Q(x). The last number is the remainder, R(x).
- Our coefficients are 1, 5, 15, and 27. Since we started with a polynomial of degree 4 (x^4), the quotient will have a degree one less, which is 3. So, Q(x) = x^3 + 5x^2 + 15x + 27.
- The remainder is 81, so R(x) = 81.
Method 2: Long Division (The Classic Approach)
Alright, let's tackle this same problem using the long division method. This approach might look a bit more involved at first, but it's super versatile and works for dividing polynomials by any other polynomial, not just linear ones.
- Set up the long division: Write P(x) (the dividend) inside the "division bracket" and D(x) (the divisor) outside. Make sure to include placeholders for any missing terms in P(x), just like we did with synthetic division. So, we have:
________________________
x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0
- Divide the leading terms: Divide the leading term of P(x) (x^4) by the leading term of D(x) (x). x^4 / x = x^3. Write x^3 above the division bracket, aligned with the x^3 term.
x^3_____________________
x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0
-
Multiply: Multiply the entire divisor D(x) (x - 3) by the term you just wrote above the bracket (x^3). x^3 * (x - 3) = x^4 - 3x^3
-
Subtract: Write the result (x^4 - 3x^3) below P(x) and subtract. Remember to change the signs of the terms you're subtracting!
x^3_____________________
x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0
- (x^4 - 3x^3)
________________________
5x^3
- Bring down the next term: Bring down the next term from P(x) (which is +0x^2) and write it next to the result of the subtraction.
x^3_____________________
x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0
- (x^4 - 3x^3)
________________________
5x^3 + 0x^2
-
Repeat: Repeat steps 2-5 until you've brought down all the terms from P(x).
- Divide the leading term of the new result (5x^3) by the leading term of D(x) (x). 5x^3 / x = 5x^2. Write +5x^2 above the bracket.
x^3 + 5x^2________________ x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0 - (x^4 - 3x^3) ________________________ 5x^3 + 0x^2- Multiply D(x) (x - 3) by 5x^2. 5x^2 * (x - 3) = 5x^3 - 15x^2
- Subtract. Be careful with the signs!
x^3 + 5x^2________________ x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0 - (x^4 - 3x^3) ________________________ 5x^3 + 0x^2 - (5x^3 - 15x^2) ________________________ 15x^2 - 18x- Bring down the next term (-18x).
- Divide 15x^2 by x. 15x^2 / x = 15x. Write +15x above the bracket.
- Multiply D(x) by 15x. 15x * (x - 3) = 15x^2 - 45x
- Subtract.
x^3 + 5x^2 + 15x____________ x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0 - (x^4 - 3x^3) ________________________ 5x^3 + 0x^2 - (5x^3 - 15x^2) ________________________ 15x^2 - 18x - (15x^2 - 45x) ________________________ 27x + 0- Bring down the last term (+0).
- Divide 27x by x. 27x / x = 27. Write +27 above the bracket.
- Multiply D(x) by 27. 27 * (x - 3) = 27x - 81
- Subtract.
x^3 + 5x^2 + 15x + 27 x - 3 | x^4 + 2x^3 + 0x^2 - 18x + 0 - (x^4 - 3x^3) ________________________ 5x^3 + 0x^2 - (5x^3 - 15x^2) ________________________ 15x^2 - 18x - (15x^2 - 45x) ________________________ 27x + 0 - (27x - 81) ________________________ 81 -
Interpret the results:
- The quotient, Q(x), is the polynomial above the division bracket: Q(x) = x^3 + 5x^2 + 15x + 27.
- The remainder, R(x), is the value left over at the bottom: R(x) = 81.
Expressing P(x) in the Form P(x) = D(x) * Q(x) + R(x)
Okay, we've done the hard work! We've divided P(x) by D(x) using both synthetic and long division (and got the same answer, which is always a good sign!). Now, let's express P(x) in the required form:
P(x) = D(x) * Q(x) + R(x)
We know:
- P(x) = x^4 + 2x^3 - 18x
- D(x) = x - 3
- Q(x) = x^3 + 5x^2 + 15x + 27
- R(x) = 81
So, plugging these in, we get:
x^4 + 2x^3 - 18x = (x - 3) * (x^3 + 5x^2 + 15x + 27) + 81
And that's it! We've successfully divided the polynomials and expressed P(x) in the desired form. You can even multiply out the right side to verify that it equals P(x) if you want to double-check your work.
Conclusion: Polynomial Division Mastered!
Guys, we did it! We successfully divided the polynomial P(x) = x^4 + 2x^3 - 18x by D(x) = x - 3 using both synthetic and long division. We then expressed P(x) in the form P(x) = D(x) * Q(x) + R(x), where Q(x) = x^3 + 5x^2 + 15x + 27 and R(x) = 81. Remember, polynomial division might seem tricky at first, but with practice, you'll become a pro! Keep practicing, and you'll be dividing polynomials like a boss in no time!