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Polynomial Arithmetic

requiresVariables & Expressions ⚠Exponents & Roots ⚠

⚗ Dr. Möbius, from the lab

Polynomials are the integers of algebra. You can add them, subtract them, multiply them, and you'll almost never divide them cleanly — just like the integers. This analogy is not decorative; it's a theorem about algebraic structure that will pay off when you meet abstract algebra. For now, master the arithmetic. Distributivity is the only engine here — everything else is just knowing when and how to apply it.

THE BIG IDEA

A polynomial is a sum of monomials; polynomial arithmetic is entirely driven by the distributive law from Bedrock.

What a polynomial is

A monomial is a product of a number (the coefficient) and non-negative integer powers of variables: $5x^3$ , $-2xy^2$ , $7$ , $x$ . A polynomial is a finite sum of monomials called terms. Examples:

$3x^2 - 7x + 4$ (degree 2, also called a quadratic)
$x^4 + 0x^3 - 2x^2 + x - 1$ (degree 4)
$5$ (degree 0 — a nonzero constant)

Standard form: terms written in decreasing order of degree. The leading term is the one of highest degree; its coefficient is the leading coefficient. The degree of a polynomial is the exponent in the leading term.

The zero polynomial ( $0$ ) has no degree by convention — we leave it undefined. This is a formal nicety; it rarely bites you.

Addition and subtraction: combine like terms

Addition and subtraction of polynomials is just combining like terms — and you now know that combining like terms IS the distributive law:

$(3x^2 - 2x + 5) + (x^2 + 4x - 3)$ $= (3+1)x^2 + (-2+4)x + (5-3) = 4x^2 + 2x + 2.$

For subtraction, distribute the negative sign first: $(3x^2 - 2x + 5) - (x^2 + 4x - 3) = 3x^2 - 2x + 5 - x^2 - 4x + 3 = 2x^2 - 6x + 8.$

That negative sign distributes as $-1$ multiplying every term in the second polynomial. Don't drop it on the last term.

Multiplication: distributivity unleashed

Multiplying two polynomials means every term of the first meets every term of the second. This is the distributive law applied repeatedly:

$(2x + 3)(x^2 - 4x + 1).$

Distribute the first factor: $= 2x(x^2 - 4x + 1) + 3(x^2 - 4x + 1)$ $= 2x^3 - 8x^2 + 2x + 3x^2 - 12x + 3$ $= 2x^3 + (-8+3)x^2 + (2-12)x + 3$ $= 2x^3 - 5x^2 - 10x + 3.$

Count: $2 \times 3 = 6$ partial products before combining. With a degree-1 times a degree-2, you get at most $1+2+1 = 4$ terms in the result.

Special products: pictures and memory

Three products appear so often they deserve their own treatment — but memorize them through the geometric pictures, not as incantations.

The square of a binomial: $(a+b)^2 = a^2 + 2ab + b^2$ .

Picture: a square of side $(a+b)$ , split into four rectangles: $a \times a$ , $a \times b$ , $b \times a$ , $b \times b$ . Clearly $a^2 + 2ab + b^2$ . The notorious student error is $(a+b)^2 = a^2 + b^2$ — this omits the two middle rectangles. Don't.

Similarly, $(a-b)^2 = a^2 - 2ab + b^2$ .

Difference of squares: $(a+b)(a-b) = a^2 - b^2$ .

The $ab$ and $-ab$ terms cancel. You'll FACTOR this pattern in the next lesson, so you need to recognize it going both directions.

The integers analogy

Polynomials with integer coefficients behave like the integers: they're closed under addition, subtraction, and multiplication, but they don't generally close under division. Dividing $x^2 + 1$ by $x-1$ doesn't give a polynomial; it gives a polynomial with a remainder, just like $7 \div 2 = 3$ remainder $1$ .

This analogy is a theorem about polynomial rings in abstract algebra. You don't need to prove it now, but notice it — every algebraic identity you find for integers tends to have a polynomial sibling.

🔬 SPECIMENS (worked examples)

Worked example 1 — addition and subtraction▸

Simplify: $(4x^3 - 2x^2 + 3x - 1) - (x^3 + 5x^2 - 2x + 4)$ .

Distribute the subtraction: $= 4x^3 - 2x^2 + 3x - 1 - x^3 - 5x^2 + 2x - 4.$

Collect like terms by degree:

$x^3$ : $4 - 1 = 3$
$x^2$ : $-2 - 5 = -7$
$x$ : $3 + 2 = 5$
constant: $-1 - 4 = -5$

Result: $3x^3 - 7x^2 + 5x - 5$ .

Degree check: we subtracted two degree-3 polynomials; the result can have degree at most 3 (and it does).

Worked example 2 — multiplication with FOIL and beyond▸

Expand $(3x - 2)(2x^2 + x - 4)$ .

Distribute each term of the first factor over the second: $3x(2x^2 + x - 4) + (-2)(2x^2 + x - 4)$ $= 6x^3 + 3x^2 - 12x - 4x^2 - 2x + 8.$

Collect like terms: $= 6x^3 + (3-4)x^2 + (-12-2)x + 8 = 6x^3 - x^2 - 14x + 8.$

Degree check: $1 + 2 = 3$ . Result has degree 3. Confirmed.

Worked example 3 — the trap: wrong special product▸

A student claims $(2x - 5)^2 = 4x^2 - 25$ . Identify the error and compute the correct expansion.

The student computed $(2x)^2 - 5^2$ , treating $(2x-5)^2$ as a difference of squares $(2x-5)(2x+5)$ . But $(2x-5)^2$ means $(2x-5)(2x-5)$ — the same factor twice.

Correct expansion using $(a-b)^2 = a^2 - 2ab + b^2$ with $a = 2x$ , $b = 5$ : $(2x-5)^2 = (2x)^2 - 2(2x)(5) + 5^2 = 4x^2 - 20x + 25.$

The missing term is $-20x$ . The middle term $-2ab$ is always there; it cannot vanish unless $a = 0$ or $b = 0$ .

Verify by direct FOIL: $(2x-5)(2x-5) = 4x^2 - 10x - 10x + 25 = 4x^2 - 20x + 25$ . Confirmed.

☠ KNOWN HAZARDS

The cardinal sin: $(a+b)^2 = a^2 + b^2$ . This is wrong. The correct expansion is $a^2 + 2ab + b^2$ . The $2ab$ is the area of the two middle rectangles in the geometric picture. Every student makes this mistake at least once; most make it repeatedly. Draw the square.
Dropping the negative on subtraction. $(a+b) - (c+d) = a+b-c-d$ , not $a+b-c+d$ . The subtraction distributes a $-1$ through the ENTIRE second polynomial.
Confusing degree of a product. The product of a degree- $m$ and degree- $n$ polynomial has degree $m+n$ , not $\max(m,n)$ . $(x^3)(x^4) = x^7$ , degree 7.
Not writing in standard form and missing like terms. After multiplying, collect terms of the same degree before reporting the answer. Leaving $3x^2 + 5x^2 - 2x$ as is instead of $8x^2 - 2x$ is an unfinished job.

TL;DR

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A polynomial is a sum of monomials; degree = the highest exponent; standard form lists terms in decreasing degree.
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Addition/subtraction: combine like terms (distributive law running backwards).
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Multiplication: every term meets every term (distributive law applied repeatedly). For two polynomials of degrees $m$ and $n$ , the product has degree $m+n$ .
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Special products: $(a+b)^2 = a^2+2ab+b^2$ , $(a-b)^2 = a^2-2ab+b^2$ , $(a+b)(a-b)=a^2-b^2$ . Know them geometrically.
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Polynomials are the integers of algebra: closed under $+$ , $-$ , $\times$ but not under $\div$ .

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Factoring →Polynomial Functions →Counting & the Binomial Theorem →