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Polynomial – Basic Formula
We will just derive some formulae that will be used over and over again in other places.
Example 1. (x+a)2 = x2 + 2ax + a2
(x+a)(x+a) = (x+a)x + (x+a)a = x2 + ax + ax + a2 = x2 + 2ax + a2
This one is useful while evaluating the square root of a number, the solution for quadratic equation, and finding the maximum or minimum values of some functions.
Example 2: (x-a)(x+a) = x2 – a2
(x-a)(x+a) = (x-a)x + (x-a)a = x2 – ax + ax – a2 = x2 – a2
This formula will help while computing (682 – 322) .
Example 3: (ax+b)(cx+d) = (ac)x2 + (bc+ad)x + bd
(ax+b)(cx+d) = (ax+b)cx + (ax+b)d = acx2 + bcx + adx + bd = acx2 + (bc+ad)x + bd
It is useful while evaluating the roots of quadratic equations while it has solutions as rational numbers. When we introduce the factorization of 2nd order polynomials, we will use it very often.
Example 4: (x-1)(x2 + x + 1) = x3 -1
(x-1)(x2 + x + 1) = x3 – x2 + x2 –x +x-1 = x3 -1
This one is useful while higher order polynomials are divided by (x2+x+1). It is easy to use this one to find the remainder polynomials in this kind of questions.
Example 5: (x+1)(x2 –x + 1) = x3 + 1
(x+1)(x2-x+1) = x3 + x2 - x2 – x +x +1 = x3 + 1
Example 6: (x-a)(x2 + ax + a2) = x3 – a3
(x-a)(x2 + ax + a2) = x3 – ax2 + ax2 –a2x +a2x – a3 = x3 – a3
Example 7: (x+a)(x2 – ax + a2) = x3 + a3
(x+a)(x2 – ax + a2) = x3 + ax2 -ax2 – a2x + a2x + a3 = x3 + a3
Example 8: (x+y+z)2 = x2+y2+z2 + 2xy+2yz+2zx
(x+y+z)2 = (x+y+z)x + (x+y+z)y + (x+y+z)z = x2 + xy + zx + xy+ y2 + zy +xz+yz+z2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
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