📋 What to do today
You'll practise dividing polynomials using the self-marking exercises below. Work steadily through Level 2, then 3, 4 and 5 — each one steps up a little.
Show all your working in your exercise book (don't just type answers), and click Check on the site as you go so you know straight away whether you're on track. Aim to get solidly through Levels 2 and 3, then push into 4 and 5.
A quick reminder first
Polynomial long division works just like the long division you did with numbers — you divide, multiply, subtract, and bring down the next term. Here's one worked through:
Worked example: $(x^2+4x+3)\div(x+1)$
x + 3
______________
x + 1 ) x^2 + 4x + 3
x^2 + x
----------
3x + 3
3x + 3
------
0
- Divide the leading terms: $x^2 \div x = x$. Write $x$ on top.
- Multiply back: $x(x+1) = x^2 + x$. Write it underneath and subtract: $(x^2+4x) - (x^2+x) = 3x$. Bring down the $+3$.
- Repeat: $3x \div x = 3$. Multiply: $3(x+1) = 3x+3$. Subtract: remainder $0$.
Answer: $x^2+4x+3 = (x+1)(x+3)$, so the quotient is $x+3$ with no remainder.
- Divide the leading term of what's left by the leading term of the divisor.
- Multiply that result by the whole divisor.
- Subtract (this is where most slips happen — watch your signs).
- Bring down the next term and repeat until nothing is left to bring down.
If there's a leftover at the end that's a lower degree than the divisor, that's your remainder — that's exactly what Level 3 is about.
Your practice — work through these in order
Divide by a linear expression
Divide a polynomial by something like $(x+1)$ or $(2x-3)$ — same as the example above.
Open Level 2 →Find the remainder
Same method, but now the division doesn't come out exactly — state the quotient and the remainder.
Open Level 3 →Divide by a quadratic or cubic
Step up to dividing by a quadratic or cubic divisor. The four steps are identical — just a bit more of it.
Open Level 4 →Mixed practice
A mixed set that pulls all of it together. Good stretch once Levels 2–4 feel comfortable.
Open Level 5 →If you get stuck
- Re-read the worked example above — most crosses come from a sign error in the subtraction step, so slow down there.
- Line your working up in columns (like terms under like terms) — it makes the subtraction far easier to get right.
- Check any answer by expanding it back out: divisor × quotient (+ remainder) should give you the original polynomial.
- Compare with someone next to you working on the same level — but write your own working.