Polynomial Review · Lesson 2
Polynomials add, subtract, and multiply by the ordinary rules of arithmetic, because a polynomial is a number in disguise. Two moves do all the work: combine like terms (for + and −) and distribute every term across every term (for ×). The single most common mistake in all of algebra lives here — the dropped minus sign — so we attack it head-on.
By the end, you can
Like terms have the exact same variable part — same variable, same exponent. 3x² and −5x² are like; 3x² and 3x are not (different exponents); 7x and −2x are like. Combine them by adding coefficients and keeping the variable part: 3x² − 5x² = −2x².
Why it works
Like terms are the same kind of thing. 3 apples plus 5 apples is 8 apples — you never turn apples into apple-squareds. The variable part is the unit; you only add the counts in front.
To add, drop the parentheses (a leading + changes nothing) and combine like terms: (3x² + 2x − 1) + (x² − 5x + 4) = 4x² − 3x + 3.
Why it works
Addition is 'pour both bags into one pile and sort.' A + in front of a parenthesis is transparent — every sign inside is unchanged.
Subtraction means add the opposite, so a minus in front of a parenthesis flips the sign of EVERY term inside, not just the first. (5x² − 3x + 2) − (2x² + 4x − 7) = 5x² − 3x + 2 − 2x² − 4x + 7 = 3x² − 7x + 9. Note the +7, from −(−7).
Why it works
The minus sign is a sign-flipping gate. Rewrite the subtraction as + (−1)·(…) and distribute the −1. The term that trips everyone is the last one: subtracting a negative makes it positive.
Multiplication is repeated distribution: multiply each term of the first factor by each term of the second, then combine like terms. For two binomials this is the four-product pattern (FOIL): (x + 3)(x + 5) = x·x + x·5 + 3·x + 3·5 = x² + 8x + 15. When multiplying variables, add exponents: x·x = x².
Why it works
Think of a rectangle with sides (x+3) and (x+5). Its area is the sum of the four sub-rectangles x·x, x·5, 3·x, 3·5. 'Every term times every term' is just adding up the pieces of the area — and it scales to any size, unlike the FOIL crutch.
Same rule with one term on the outside: distribute it to each inside term. −2x(3x² − 4x + 1) = −6x³ + 8x² − 2x. Watch the sign AND the exponent on every product.
Why it works
One hand hands the outside factor to each term inside, one at a time. Skipping a term is like shortchanging one person in the room.
Combine: 4x + 9 − 7x + 2.
Group like terms
(4x − 7x) + (9 + 2).
Add the coefficients
4 − 7 = −3; 9 + 2 = 11.
Add (2x² + x) + (3x² − 4x).
Collect x² terms
2x² + 3x² = 5x².
Collect x terms
x − 4x = −3x.
Subtract (6x² + 2x − 5) − (x² − 3x − 8).
Flip every sign inside
6x² + 2x − 5 − x² + 3x + 8. Watch: −(−8) = +8.
Combine like terms
(6−1)x² + (2+3)x + (−5+8) = 5x² + 5x + 3.
Multiply 3x(2x² − 5x + 4).
Distribute 3x to each term
3x·2x² = 6x³; 3x·(−5x) = −15x²; 3x·4 = 12x.
Expand (2x − 3)(x + 4).
Four products
2x·x = 2x²; 2x·4 = 8x; −3·x = −3x; −3·4 = −12.
Combine the middle
8x − 3x = 5x.
Expand (x + 2)(x² − 3x + 5).
Distribute x
x³ − 3x² + 5x.
Distribute 2
2x² − 6x + 10.
Add, lining up by degree
x³ + (−3+2)x² + (5−6)x + 10 = x³ − x² − x + 10.
Simplify 2(x² − x) − 3(2x² + x − 4).
Distribute both
2x² − 2x and −6x² − 3x + 12. Note −3·(−4) = +12.
Combine like terms
(2−6)x² + (−2−3)x + 12 = −4x² − 5x + 12.
Distributing a subtraction to only the first term: (…) − (2x² + 4x − 7) = … − 2x² + 4x − 7.
Wrong. The minus flips EVERY sign: −2x² − 4x + 7. The last term especially: −(−7) = +7. Rewrite as + (−1)·(2x² + 4x − 7) if it helps.
Combining unlike terms: 3x² + 4x = 7x³ or 7x².
You cannot merge different powers. 3x² + 4x is already simplified — they are different 'kinds of thing'.
Adding exponents when ADDING terms: x² + x² = x⁴.
Adding like terms adds the coefficients: x² + x² = 2x². Exponents add only when you MULTIPLY: x²·x² = x⁴.
Multiplying only the 'outer' terms of two binomials and forgetting the middle.
Every term times every term. For two binomials that's four products, not two — don't drop the inner and outer cross-terms.
14 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.
t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).Combine: 8x − 3 + 2x + 10.
Add: (x² + 4x − 6) + (2x² − x + 1).
Subtract: (5x − 2) − (3x + 7).
Multiply: 4x(x² − 2x + 3).
Expand: (x + 6)(x + 2).
Expand: (2x − 3)(x + 4).
Subtract: (6x² + 2x − 5) − (x² − 3x − 8).
Expand: (2x − 5)(3x + 1).
Simplify: 5(x² + 2x) − 2(x² − 3x + 4).
Expand: (x + 2)(x² − 3x + 5).
Which statement is correct?
Expand and simplify (x − 1)(x² + x + 1).
Find P if P − (x² − 3x) = 2x² + x − 5.
What is the most common mistake in computing (…) − (2x² + 4x − 7)?