How this instrument works
Add two polynomials by adding the coefficients that sit on the same power of x, and nothing else: the x² coefficients combine with each other, the x coefficients combine with each other, and the constants combine with each other. Nothing here multiplies terms of different degree together — that is multiplication's job, not addition's. The grouping is legal because addition is associative and commutative: (a1x² + b1x + c1) + (a2x² + b2x + c2) can be reordered freely, so the x² terms drift together, the x terms drift together, and (a1 + a2)x² falls out by running the distributive law in reverse.
Subtraction is the same grouping with one extra step: every coefficient of the second polynomial changes sign before it joins its partner, because P1 − P2 means P1 + (−1)·P2, and multiplying by −1 flips the sign on all three terms, not only the one written first. This is exactly where marks get lost — a subtraction that flips the leading x² coefficient but forgets the constant term is not really a subtraction; it is subtracting one term and adding two others by accident.
One consequence worth sitting with: the degree of a sum or difference is not fixed by the degree of the inputs. Two quadratics can add to a quadratic, as they do below, but if their x² coefficients happened to be equal and opposite, that leading term would cancel and the result would collapse to a straight line — degree one, not two. Tracking each power separately, rather than reading a polynomial as one indivisible number, is the same habit the word 'algebra' was built on: al-Khwarizmi's al-jabr meant restoring balance by moving and combining terms, which is exactly what a coefficient-by-coefficient sum or difference does.
- Type P₁'s coefficients into P₁: x² coefficient, P₁: x coefficient, and P₁: constant.
- Type P₂'s coefficients into P₂: x² coefficient, P₂: x coefficient, and P₂: constant.
- Read the sum in Sum: x² coefficient, Sum: x coefficient, and Sum: constant, for P₁ + P₂.
- Read the difference in Difference: x² coefficient, Difference: x coefficient, and Difference: constant, for P₁ − P₂.
- Adjust any of the six coefficients and every result updates together, since the sum and difference share the same inputs.
Worked example — combining 2x² + 3x + 4 and x² − x + 2
Let P₁ = 2x² + 3x + 4 and P₂ = x² − x + 2 — enter a1=2, b1=3, c1=4 and a2=1, b2=−1, c2=2. The sum adds each pair of like terms: x² coefficients 2+1=3, x coefficients 3+(−1)=2, constants 4+2=6, so P₁+P₂ = 3x²+2x+6 — the Sum fields read sumA=3, sumB=2, sumC=6.
The difference flips every sign in P₂ before combining: x² coefficients 2−1=1, x coefficients 3−(−1)=4 — subtracting a negative adds — and constants 4−2=2, so P₁−P₂ = x²+4x+2 — the Difference fields read diffA=1, diffB=4, diffC=2. Notice the middle term jumps from 2 in the sum to 4 in the difference, purely from that one sign flip.
Questions
How do you add two polynomials?
Add the coefficients that share the same power of x and leave every other coefficient alone: x² terms combine with x² terms, x terms with x terms, constants with constants. For 2x²+3x+4 and x²−x+2 that gives (2+1)x²+(3−1)x+(4+2) = 3x²+2x+6 — three independent additions, not one big one.
Why does subtracting a polynomial flip every sign, not just the first term?
Because P₁−P₂ means P₁+(−1)·P₂, and distributing that −1 multiplies all three terms of P₂ by −1, not only the leading one. Forgetting to flip the constant term is the single most common slip — in the worked example the constant goes from +2 in P₂ to −2 inside the subtraction, giving 4−2=2, not 4+2.
Can adding two quadratics produce a polynomial of lower degree?
Yes — if the x² coefficients are equal and opposite, they cancel and the sum collapses to a linear expression or lower. The reverse never happens: the degree of a sum or difference can drop but it can never rise above the highest degree already present in the inputs, since no new powers of x are created by adding or subtracting.
Is adding polynomials the same operation as adding ordinary numbers?
It is the same law of addition applied three times in parallel rather than once. A number is really a polynomial with only a constant term, so adding two numbers is the c1+c2 step alone; adding full quadratics runs that identical addition once per power of x, for x², x, and the constant, independently of each other.
Does the order matter — is P₁−P₂ the same as P₂−P₁?
No, and that is the key difference from addition. P₁+P₂ always equals P₂+P₁, but P₁−P₂ and P₂−P₁ are negatives of each other, term by term: here P₁−P₂ = x²+4x+2, while P₂−P₁ would be −x²−4x−2 — every coefficient's sign reversed, not the expression rearranged.
Where does the term 'like terms' come from?
Like terms are built from the same power of the variable — 3x² and −5x² are like terms, but 3x² and 3x are not, however equal their coefficients look. Only like terms combine by addition or subtraction; unlike terms sit side by side in the answer, which is why a quadratic sum still carries three separate coefficients rather than one merged number.