Testwiki:Reference desk/Archives/Mathematics/2017 March 12

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March 12

Formula for product of x-intercepts (if all real numbers) of a polynomial

Is there any formula for the product of all x-intercepts of a polynomial?? A linear polynomial always has -b/m as its sole x-intercept. A quadratic polynomial with 2 x-intercepts will have them multiply to -c/a. Is there any formula for the product of all x-intercepts of any polynomial?? Georgia guy (talk) 19:49, 12 March 2017 (UTC)

$A\prod _{j=1}^n(x-{\alpha }_j)=A{x}^n+\cdots +(-1{)}^{n}A\prod _{j=1}^n{\alpha }_j$ Count Iblis (talk) 20:06, 12 March 2017 (UTC)

Template:Ping More generally, see Vieta's formulas. The product of the roots is always equal to the quotient of the coefficient of the constant term by the coefficient of the highest-order nonzero term, times -1 raised to the power of the degree of the polynomial.

This includes complex roots as well. In general, there is no easy way to remove the complex roots from the product and still have a clean formula in terms of the coefficients.--Jasper Deng (talk) 20:11, 12 March 2017 (UTC)