Trinomial

derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.

derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial ]] In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. ==Examples of trinomial expressions== 3x + 5y + 8z with x, y, z variables 3t + 9s^2 + 3y^3 with t, s, y variables 3ts + 9t + 5s with t, s variables ax^2+bx+c , the quadratic polynomial in standard form with a,b,c variables. A x^a y^b z^c + B t + C s with x, y, z, t, s variables, a, b, c nonnegative integers and A, B, C any constants. Px^a + Qx^b + Rx^c where x is variable and constants a, b, c are nonnegative integers and P, Q, R any constants. ==Trinomial equation== A trinomial equation is a polynomial equation involving three terms. An example is the equation x = q + x^m studied by Johann Heinrich Lambert in the 18th century. a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) A special type of trinomial can be factored in a manner similar to quadratics since it can be viewed as a quadratic in a new variable ( below). This form is factored as: x^{2n} + rx^n + s = (x^n + a_1)(x^n + a_2), where \begin{align} a_1+a_2 &= r\\ a_1 \cdot a_2 &= s. \end{align} For instance, the polynomial is an example of this type of trinomial with . The solution and of the above system gives the trinomial factorization: . The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process. ==See also== Pascal%27s_pyramid Trinomial expansion Monomial Binomial Multinomial Simple expression Compound expression Sparse polynomial ==Notes== ==References==

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