Operate on polynomials and prove identities

Try

A model uses P(x)=x²+2x+3 and Q(x)=2x²−x+1. To combine them, first organize coefficients by power. A polynomial is a finite sum of terms aᵢx^i, where each exponent i is a nonnegative integer and each coefficient aᵢ is real. Missing powers have coefficient zero. This organization makes subtraction safer than trying to read all the signs at once.

Predict

Predict the degree of P+Q and PQ. Could cancellation make a sum have smaller degree? Could multiplying two nonzero real polynomials make the highest-degree term disappear? Explain your predictions using leading coefficients. Also predict whether dividing one polynomial by another always gives a polynomial; the small example 1 divided by x is a useful test.

Experiment

Construct a coefficient table, including zeros when necessary. Here is the completed addition/subtraction table; redo it with Q−P to check that subtraction order matters.

Power	P coefficient	Q coefficient	P+Q	P−Q
x²	1	2	3	−1
x	2	−1	1	3
1	3	1	4	2

For multiplication, make a grid with the three P terms along one side and the three Q terms along the other. Put their product in each of the nine cells, then combine cells with the same total exponent. The result is 2x⁴+3x³+5x²−x+3. At x=1 the original product is 6·2=12, matching the expanded expression. This is a useful error check, not a proof of identity at every input.

Understand

Addition combines corresponding coefficients. Subtraction changes the sign of every term in the subtracted polynomial, including its constant. Multiplication distributes every term against every term: aᵢx^i times bⱼx^j contributes aᵢbⱼ to the coefficient of x^(i+j). The exponent addition explains why the multiplication grid is organized along equal-degree diagonals.

These operations are closed within real polynomials. There are only finitely many input terms, so only finitely many resulting terms; sums and products of real coefficients remain real; and adding nonnegative integer exponents cannot produce a negative or fractional exponent. The zero polynomial can result from cancellation. A nonzero product has degree equal to the sum of degrees because the product of its nonzero leading coefficients is nonzero. Division has no such closure: 1/x is undefined at zero and has a negative power, so it is not a polynomial.

An identity is a statement valid for all permitted inputs. For example, distribution proves (u−v)(u+v)=u²+uv−uv−v²=u²−v². It follows that 97·103=(100−3)(100+3)=10000−9=9991. The symbolic argument proves the general relationship; the numerical calculation uses it.

A second useful identity is (u²+v²)²=(u²−v²)²+(2uv)². Expansion on the right combines −2u²v² with 4u²v², leaving the positive middle term in the left square. For u=3,v=2 it gives legs 5 and 12 and hypotenuse 13. This generates a numerical relationship without assuming every possible triple has been classified.

Apply

The practice includes all three operations, a closure explanation and two identity applications. For a product, inspect one coefficient through all of its contributing cells; for a difference, check every changed sign. To prove an identity, use a general operation law or coefficient argument, then apply it to an unfamiliar numerical example.

Do not confuse an equation such as x²=x, which holds only at particular inputs, with an identity. Finding one counterexample refutes a universal identity claim, while finding one successful input does not prove it.

Continue

Polynomial division and the binomial theorem already have separate lessons. Reuse those when needed; the next lesson focuses instead on rational-expression arithmetic, where polynomial operations still work but input restrictions and nonzero divisors become essential.

Continue to Keep domains through rational arithmetic.