A polynomial is a fundamental concept for the solution of algebraic equations and representations of exponential, rational and other functions. This structure is the most common in school course subject to the square ofthe nth equation.
Often as simplifying the cumbersome expressions, there is a need to build a trinomial into the square. This is no ready-made formula, but there are several methods. For example, imagine a square trinomialand the product of two identical terms.
Consider this example: construct in square trinomial 3•x2 + 4•x – 8.
Change the entry (3•x2 + 4•x – 8)2 (3•x2 + 4•x – 8)•( 3•x2 + 4•x – 8) and use the rule of multiplication of polynomials, which consists in sequential computation works. First, multiply the first component of the first parentheses to each term of the second, then do the same with the second and finally the third:(3•x2 + 4•x – 8)•( 3•x2 + 4•x – 8) = 3•x2•(3•x2 + 4•x - 8) + 4•x•(3•x2 + 4•x – 8) – 8•(3•x2 + 4•x – 8) = 9•x^4 + 12•X3 – 24•x2 + 12•X3 + 16•x2 – 32•x – 24•x2 – 32•x + 64 = 9•x^4 + 24•X3 – 32•x2 – 64•x + 64.
The same result can come, if we remember that the result of multiplying two trinomials remains the sum of six items, three of which are square, AMI each term, and the other three are their various pairwise works in the doubled form. This simple elementary formula looks like this:(a + b + c)2 = a2 + b2 + c2 + 2•a•b + 2•a•c + 2•b•c.
Apply it to your example:(3•x2 + 4•x - 8)2 = (3•x2 + 4•x + (-8))2 =(3•x2)2 + (4•x)2 + (-8)2 + 2•(3•x2)•(4•x) + 2•(3•x2)•(-8) + 2•(4•x)•(-8) = 9•x^4 + 16•x2 + 64 + 24•X3 – 48•x2 – 64•x = 9•x^4 + 24•X3 - 32•x2 - 64•x + 64.
As you can see, the answer was the same, but the manipulation took less than. This is especially important when the terms are themselves complex structures. This method is applicable for the trinomialand any degree and any number of variables.