What Is 3x³ - 11x² - 26x + 30 Divided by x - 5?
So you've got this polynomial division problem staring back at you: (3x³ - 11x² - 26x + 30) ÷ (x - 5). Worth adding: maybe it's homework. Maybe you're reviewing for a test. Either way, you need the answer and — more importantly — you want to understand how to get there That's the whole idea..
Not obvious, but once you see it — you'll see it everywhere.
Here's the short version: 3x³ - 11x² - 26x + 30 divided by x - 5 equals 3x² + 4x - 6, with no remainder. But let me walk you through exactly how we get there, because the process matters just as much as the answer The details matter here..
What Is Polynomial Division?
Polynomial division is basically long division, but with variables instead of just numbers. You're dividing a polynomial (a fancy expression with multiple terms like x³, x², x, and constants) by another polynomial — in this case, a simple linear one: x - 5 And that's really what it comes down to..
If you're divide polynomials, you're looking for two things:
- The quotient — the answer you'll get (in this case, another polynomial)
- The remainder — what's left over, if anything
For this particular problem, you'll end up with a clean quotient and zero remainder. That means x - 5 is actually a factor of the original polynomial — it divides in perfectly.
Why Synthetic Division Works Faster
Here's a pro tip: when you're dividing by something in the form x - c (where c is just a number), you can skip the lengthy long division process and use synthetic division instead. It's faster, cleaner, and way less prone to small arithmetic errors.
For dividing by x - 5, we use c = 5 in our synthetic division setup.
How to Solve It (Step by Step)
Let's work through this together. I'll show you the synthetic division method since it's the quickest path to the answer.
Step 1: Set Up Your Coefficients
First, identify every coefficient in the dividend (the polynomial you're dividing):
- 3x³ → coefficient 3
- -11x² → coefficient -11
- -26x → coefficient -26
- 30 → coefficient 30
Write these in order: 3, -11, -26, 30
Step 2: Set Up the Synthetic Division Box
Draw a small box with a line at the top. Day to day, on the right side of that line, write the value from our divisor. Since we're dividing by x - 5, we use 5.
Then write your coefficients to the left of the vertical line:
5 | 3 -11 -26 30
Step 3: Bring Down the First Coefficient
Bring the 3 straight down below the line. This becomes the first number in your answer.
5 | 3 -11 -26 30
| 3
Step 4: Multiply, Add, Repeat
This is where the magic happens. You work from left to right:
- Multiply 5 by the 3 you just brought down → 15
- Add that to -11 → -11 + 15 = 4
- Write 4 below the line
Now multiply 5 by that 4 → 20 Add that to -26 → -26 + 20 = -6 Write -6 below the line
Finally, multiply 5 by -6 → -30 Add that to 30 → 30 + (-30) = 0 Write 0 below the line
Your bottom row should read: 3, 4, -6, 0
Step 5: Read Your Answer
Those bottom numbers tell you everything. The last number is your remainder — 0, which means it divides perfectly with nothing left over Simple, but easy to overlook..
The other numbers become your quotient. Since we started with an x³ term, our quotient will be one degree lower: x².
So: 3x² + 4x - 6
That's your answer.
Step 6: Verify (Because Always Check Your Work)
Multiply your quotient by the divisor:
(x - 5)(3x² + 4x - 6)
= x(3x² + 4x - 6) - 5(3x² + 4x - 6)
= 3x³ + 4x² - 6x - 15x² - 20x + 30
= 3x³ - 11x² - 26x + 30 ✓
It matches exactly. Your answer is correct.
Why It Matters
You might be wondering why this matters beyond passing a test. Here's the thing: polynomial division shows up in some surprising places.
In algebra, it's how you factor polynomials — finding the roots (where the graph crosses the x-axis) essentially requires dividing and testing different divisors. In calculus, you'll need it for partial fraction decomposition, which shows up when integrating rational functions. And in higher math, polynomial division is foundational for understanding things like minimal polynomials and algebraic structures.
But even if you're just trying to pass Algebra II, knowing how to divide polynomials cleanly gives you a huge advantage. It saves time on tests, reduces errors, and — honestly — just feels satisfying when you get a nice clean answer like this one with zero remainder Which is the point..
Common Mistakes People Make
Let me save you some pain here. These are the errors I see most often:
Forgetting to include zero coefficients. If a polynomial is missing an x term (like 3x³ - 11x² + 30, with no x term), you still need to write a 0 for that spot. Skipping it throws off your entire alignment Easy to understand, harder to ignore. Still holds up..
Using the wrong sign on the divisor. For x - 5, you use 5 in synthetic division. For x + 5, you'd use -5. The sign flips. This trips up a lot of people.
Not writing all terms in the quotient. The quotient's degree is always one less than the dividend's. Since we started with x³, our answer is x² + something. Don't stop at just "3" because that's only the first term.
Rushing the arithmetic. This is just regular addition and multiplication, but one small mistake in the middle ruins everything. Take your time on each step.
Practical Tips for Polynomial Division
A few things that actually help:
- Always write every coefficient, even the zeros. It keeps everything aligned.
- Check your remainder at the end. If it's not zero, you either made an arithmetic error or the divisor isn't a factor of the polynomial.
- Use synthetic division whenever the divisor is x ± a number. Save long division for messier divisors like x² + 3 or x² - x + 1.
- Verify by multiplication if there's any doubt. It takes 10 seconds and confirms you got it right.
FAQ
What is 3x³ - 11x² - 26x + 30 divided by x - 5?
The answer is 3x² + 4x - 6 with a remainder of 0.
How do you divide polynomials by x - 5?
Use synthetic division with c = 5. Write the coefficients of the polynomial (3, -11, -26, 30), bring down the first number, multiply by 5, add to the next coefficient, and repeat until you've gone through all terms.
What does it mean when the remainder is 0?
It means x - 5 is a factor of the polynomial. The polynomial can be written as (x - 5)(3x² + 4x - 6).
Can you solve this with long division instead?
Yes. That said, you can set up the long division with x - 5 outside the bracket and 3x³ - 11x² - 26x + 30 inside, then divide term by term. Still, synthetic division is just a shortcut for divisors in the form x - c. You'd get the same answer: 3x² + 4x - 6 The details matter here..
How do you verify your answer?
Multiply the quotient (3x² + 4x - 6) by the divisor (x - 5). You should get the original polynomial: 3x³ - 11x² - 26x + 30.
The Bottom Line
Polynomial division isn't magic — it's just a structured process. Once you know the steps, you can work through any problem like this one. And this particular problem happens to have a clean, satisfying answer: 3x² + 4x - 6.
If you're working through similar problems, the same method applies every time. Consider this: coefficients in a row, bring down the first one, multiply and add, repeat until you're done. The pattern never changes — only the numbers do It's one of those things that adds up. No workaround needed..
