The formula this page uses
aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, with the exponents strictly decreasing
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Standard Form Polynomial Calculator
Solve standard form polynomial problems with clear steps, notation, and a final check.
x²
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Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Algebra
Standard Form Polynomial Calculator explained
The short version
- Standard form lists a polynomial's terms from the highest power down to the constant.
- Once it is sorted, the degree, the leading coefficient and the y-intercept are all sitting where you can read them.
- Getting there means expanding every bracket and combining every pair of like terms first.
What each part means
SymbolWhat it means
aₙ — Leading coefficientThe number on the highest-power term. Its sign controls the right-hand end of the graph.
n — DegreeThe exponent on the first term once sorted.
a₀ — Constant termThe last term, with no variable. It equals the y-intercept.
aᵢ — Missing coefficientsA gap counts as 0, which matters when you set up synthetic division.
Show your work: a full example
- The expression as given7 − 2x³ + x − 5x²
- Tag each term with its degree7 is degree 0, −2x³ is degree 3, x is degree 1, −5x² is degree 2
- Sort from highest degree down−2x³ − 5x² + x + 7
- Read the degree3
- Read the leading coefficient with its sign−2
- Read the y-interceptthe constant 7, so the curve passes (0, 7)
- Predict the endsodd degree with a negative lead, so it rises on the left and falls on the right
A second, different case
- A different case: expand a product first(2x − 3)(x² + 4x − 1)
- Multiply 2x through2x³ + 8x² − 2x
- Multiply −3 through−3x² − 12x + 3
- Write everything together2x³ + 8x² − 2x − 3x² − 12x + 3
- Combine the x² terms8x² − 3x² = 5x²
- Combine the x terms−2x − 12x = −14x
- Standard form2x³ + 5x² − 14x + 3, and testing x = 1 gives 2 + 5 − 14 + 3 = −4, matching (2 − 3)(1 + 4 − 1) = −4
Copy-ready example
7 − 2x³ + x − 5x²
The expression as given
Questions about the Standard Form Polynomial Calculator
Why does standard form matter if the polynomial is the same either way?
Because three facts become free to read: the first term gives degree and end behaviour, and the last gives the y-intercept. Synthetic division, long division and most graphing routines also assume the terms arrive in that order.
Can standard form have fractions or decimals in it?
Yes. ½x² − 0.4x + 3 is perfectly standard, because the rule only constrains the order of the exponents. Many teachers ask you to clear fractions afterwards, but that is a tidiness preference, not a definition.
What do I do when two terms have the same power?
Add their coefficients into a single term, because standard form allows exactly one term per exponent. 8x² − 3x² becomes 5x², and if they cancel to zero the term disappears and the degree may drop.
What happens to a power that is missing?
Its coefficient is 0 and the term is simply not written. That is fine on paper but not in synthetic division, where you must insert a 0 placeholder or every later column shifts and the answer comes out wrong.