Numbers POLYNOMIAL: Coefficient Order, Missing Terms and Worked Models

• Introduced: The original introduction release and date are not established by the checked references.
• Availability: Currently documented in Apple’s Functions Help 15.4 documentation edition; no final supported-version endpoint is established.
• Scope: Shared iWork reference. Function-specific historical availability across Mac, iPad, iPhone and iCloud is unverified.
• Checked: October 14, 2026, against the official POLYNOMIAL reference. Examples were independently calculated; no native Numbers test was performed.

POLYNOMIAL(x-value, coefficients)

Both arguments are required. The first is a numeric input and the second is a collection of numeric coefficients. Four coefficients describe powers 3,2,1,0, in that order. A missing term needs an explicit zero coefficient so the remaining powers retain their positions.

The official POLYNOMIAL reference uses consecutive nonnegative integer powers. It directs readers to SERIESSUM when they need a different initial power or step between exponents. Neither function fits coefficients to observations merely because the input is a table of measured data.

Create a Numbers table named CubicValues, with eight columns A:H and ten rows, including one header row. Enter Coefficient in A1, Reversed in B1, x in C1, Polynomial in D1, Expanded in E1 and Difference in F1.

Put 2,-3,0,5 down A2:A5. Enter -2,-1,0,1,2,3 down C2:C7. Keep the coefficient cells numeric, including the zero in A4.

In D2 enter =POLYNOMIAL(C2,$A$2:$A$5) and fill through D7. In E2 enter =2*C2^3-3*C2^2+5 and fill through E7. Finally put =D2-E2 in F2 and fill through F7.

The dollar signs keep the coefficient range fixed while x changes. At zero, only the constant remains. A zero result at x = -1 shows that this particular input is a root; POLYNOMIAL has evaluated a supplied point, rather than searched for every root.

For the x = 2 row, the terms are 16,-12,0,5, totaling nine. The POWER function can express the individual powers too; the exponent operator makes their order especially visible here.

Put these direct controls in H2:H4:

H2 and H3 reproduce Apple’s constant-polynomial examples. A single coefficient has power zero, so it is the constant itself, not a multiplier of x.

Now enter 5,0,-3,2 down B2:B5. Put =POLYNOMIAL(2,B2:B5) in H5; expect 36. Reversing the original list changed the expression to 5*x^3-3*x+2.

Enter the label Zero omitted in B6, and 2,-3,5 in B7:B9. In H6 enter =POLYNOMIAL(2,B7:B9) for 7. Those three coefficients define a quadratic, 2*x^2-3*x+5; deleting the original zero did not merely remove a harmless empty term.

For a carefully scoped comparison, put =SERIESSUM(2,0,1,B2:B5) in H7 for 9. With starting power zero and step one, SERIESSUM reads the reversed list as constant, linear, quadratic and cubic coefficients. Its official reference specifies that sequence. Do not transfer a coefficient list between the two functions without checking the exponent order.

Suppose a demonstration model is y=0.02*x^2+1.5*x+3. Its coefficients are supplied here; they have not been estimated from the comparison readings.

Create ModelCheck, with ten columns A:J and nine rows, including one header row. Put 0.02,1.5,3 down A2:A4 and 0,10,20,30,40 down C2:C6. In D2 enter =POLYNOMIAL(C2,$A$2:$A$4) and fill through D6.

In E2 enter =(0.02*C2+1.5)*C2+3 and fill through E6. This nested multiplication is an independent algebraic reconstruction, not a claim about Apple’s internal implementation. Put =D2-E2 in F2 and fill through F6; all five differences should be zero for the mathematical model.

Enter the synthetic comparison readings 3.2,19.7,41.4,65.8,95.1 down G2:G6. In H2 enter =G2-D2 and fill through H6:

In J2 enter =SUM(H2:H6) for 0.2. In J3 enter =H2^2+H3^2+H4^2+H5^2+H6^2 for 0.34. Positive and negative residuals partly cancel in J2, so a small signed total is not a complete measure of agreement.

In I3 enter =(D3-D2)/(C3-C2) and fill through I6. The average change per input unit over successive ten-unit intervals is 1.7,2.1,2.5,2.9. These are interval slopes, not a claim that the curve has one constant slope.

Finally, =POLYNOMIAL(50,A2:A4) in J4 gives 128. The arithmetic extends beyond the displayed range, but these invented readings provide no empirical validation of extrapolation or of a real calibration model.

• Wrong answer after copying: Anchor the coefficient range and let only the input reference move.
• Wrong degree or term order: Include internal zero coefficients and start with the highest power. Do not sort coefficients by their numeric magnitude.
• A coefficient cell is blank or text: The documented input is numeric. Empty-cell coercion, imported strings and exact error messages are not established by these examples.
• Large powers or nearly cancelling terms: Keep full precision, and do not infer a precision guarantee or overflow limit from these small exact-model checks.

SERIESSUM exposes the initial power and exponent step. POWER evaluates an individual power, and SUM can combine separately calculated terms. All serve different roles from finding unknown coefficients or solving for an unknown x.