What are the Special Products of Polynomials?

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Answer 1 · 2015-01-05T12:46:04

The general form for multiplying two binomials is:
$(x + a) (x + b) = x^{2} + (a + b) x + a b$ $(x+a)(x+b)=x^2+(a+b)x+ab$

Special products:

the two numbers are equal, so it's a square:
$(x + a) (x + a) = {(x + a)}^{2} = x^{2} + 2 a x + a^{2}$ $(x+a)(x+a)=(x+a)^2=x^2+2ax+a^2$ , or
$(x - a) (x - a) = {(x - a)}^{2} = x^{2} - 2 a x + a^{2}$ $(x-a)(x-a)=(x-a)^2=x^2-2ax+a^2$
Example : ${(x + 1)}^{2} = x^{2} + 2 x + 1$ $(x+1)^2=x^2+2x+1$
Or: $51^{2} = {(50 + 1)}^{2} = 50^{2} + 2 \cdot 50 + 1 = 2601$ $51^2=(50+1)^2=50^2+2*50+1=2601$
the two numbers are equal, and opposite sign:
$(x + a) (x - a) = x^{2} - a^{2}$ $(x+a)(x-a)=x^2-a^2$
Example : $(x + 1) (x - 1) = x^{2} - 1$ $(x+1)(x-1)=x^2-1$
Or: $51 \cdot 49 = (50 + 1) (50 - 1) = 50^{2} - 1 = 2499$ $51*49=(50+1)(50-1)=50^2-1=2499$