How do I add or divide polynomials?

How it works

Every coefficient is an exact fraction, so there is no rounding in the algebra. Each polynomial is multiplied out and its like terms are added before the operation.

• p + q, p − q, p × q: term by term, like terms added.
• p ÷ q: polynomial long division. While what is left has degree at least deg q, divide its leading coefficient by the leading coefficient of q, subtract that multiple of q times the right power of the letter, and add the term to the quotient. The result satisfies p = q × quotient + remainder with deg(remainder) less than deg(q). Division by q = 0 gives a message.
• Zeros of p: the distinct real zeros. The page takes the square-free part s = p ÷ gcd(p, p′) (exact), whose zeros are those of p, each once. While s has degree 2 or more, it finds the complex roots of s numerically (Durand-Kerner iteration, 800 rounds, starting on a circle whose radius is Fujiwara’s bound 2 × max |aₙ₋ₖ/aₙ|^(1/k), with a₀ halved). For each root whose imaginary part is within 10⁻⁷ × max(1, |real part|) of 0, it tries in turn the convergents of the continued fraction of the real part that lie within 10⁻³ × max(1, |root|) of it, with denominators up to 10¹²; a convergent r is a zero when s divided by (x − r) leaves remainder exactly 0. Then s is divided by (x − r) and the roots are found again. When s has degree 1, its zero −s₀/s₁ is a fraction. When a round finds no fraction, each real root left is a decimal after 5 Newton steps on s, to 10 significant figures, marked ≈, but only when its error bound (|s(x)| + 10⁻¹⁵ × the sum of the sizes of the terms of s) / |s′(x)| is at most 10⁻¹³ × max(1, |x|); otherwise the page says "No verified answer: some real zeros could not be confirmed to 10 significant figures (they are very close together, or very sensitive to rounding)." Last, the page counts the distinct real zeros of p exactly by Sturm’s theorem: for p, p′ and the exact remainders rᵢ of the Euclidean algorithm on them, the count is V(−∞) − V(+∞), where V is the number of sign changes of (−1)^⌊i/2⌋ rᵢ. If the count differs from the number of zeros found (zeros closer together than the numeric roots can separate look complex), the page gives the same message. Degrees up to 20.

What you can type

• One letter, the same in p and q: any lowercase letter except e. The constants e and π are not allowed, as a letter or in a coefficient.
• Numbers can have decimals (2.5 is 5/2) and powers of ten (1e-3). Fractions such as 3/4 are exact.
• Operations: + − * / and ^ with whole-number powers (a power of the letter up to 100, a bracket with a sum up to the 20th power); brackets group; numbers and letters side by side multiply (2x, 3(x + 1)). Dividing by a number is fine; dividing by an expression with the letter is not a polynomial.

How answers are written

• Result and Remainder: polynomials in the syntax you type, as MathML, highest power first: 2x^2 + 5x − 7. A coefficient that is a fraction is written over the term: 5x/6 means (5/6)x, and 3x^2/4 means (3/4)x². The remainder of an exact division is 0.
• Real zeros: "x = −1/2, x = 1, x ≈ 1.414213562" in increasing order, fractions exact (in lowest terms), decimals to 10 significant figures after ≈; "No real zeros" when there are none. Complex zeros are not listed.

Assumptions

• Coefficients are exact fractions: 0.1 is 1/10.
• A zero that is not a fraction is a decimal to 10 significant figures.

Worked examples by hand

(6x³ + 11x² − 31x + 15) ÷ (3x − 2) (OpenStax College Algebra 2e, section 5.4, Example 2). 6x³ ÷ 3x = 2x²; subtract 2x²(3x − 2) = 6x³ − 4x² to leave 15x² − 31x + 15. 15x² ÷ 3x = 5x; subtract 15x² − 10x to leave −21x + 15. −21x ÷ 3x = −7; subtract −21x + 14 to leave 1. Quotient 2x² + 5x − 7, remainder 1.

(2x + 1)(3x² − x + 4) (section 1.4, Example 4). 2x(3x² − x + 4) + 1(3x² − x + 4) = 6x³ − 2x² + 8x + 3x² − x + 4 = 6x³ + x² + 7x + 4.

Zeros of 4x³ − 3x − 1 (section 5.5, Example 5). x = 1 is a zero: 4 − 3 − 1 = 0. Dividing by x − 1 gives 4x² + 4x + 1 = (2x + 1)², so the other zero is −1/2 (twice). The page lists x = −1/2, x = 1.