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The non-real factors come in pairs which when multiplied give quadratic polynomials with real coefficients. Since every polynomial with complex coefficients can be factored into 1st-degree factors (that is one way of stating the fundamental theorem of algebra ), it follows that every polynomial with real coefficients can be factored into ...
Deduce the candidate of zero of the polynomial from its leading coefficient and constant term . (See Rational Root Theorem .) Use the factor theorem to conclude that ( x − a ) {\displaystyle (x-a)} is a factor of f ( x ) {\displaystyle f(x)} .
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
For univariate polynomials with complex coefficients, factorization can easily be reduced to numerical computation of polynomial roots and multiplicities. In the multivariate case, a random infinitesimal perturbation of the coefficients produces with probability one an irreducible polynomial, even when starting from a polynomial with many factors.
Every polynomial with rational coefficients, may be factorized, in a unique way, as the product of a rational number and a polynomial with integer coefficients, which is primitive (that is, the greatest common divisor of the coefficients is 1), and has a positive leading coefficient (coefficient of the term of the highest degree). For example:
The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term. [2] The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its ...
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