First, by
FOILing (x+10)(x+10) = x^2 + 20x + 100 which is NOT the same as the original trinomial polynomial
Next, and most importantly,
Remember the rule for factoring...
if the constant term is positive, the factors have the SAME sign which add up to the cross term in the middle
if the constant term is negative, the factors have different signs, which when added have a NET result
in the cross term of the polynomial.
X^2 + 100 = X^2 + 0x + 100
= ( x + )(x + ) <---- the only way to get x^2 is x times x;
the constant term is positive, so the signs must be the same
Now we need two numbers that multiply to 100 and ADD up to zero.
1 x 100
2 x 50
4 x 25
5 x 20
10 x10
None of them work!!! So it can't be factored.
Finally, you can test it using the discrimminant part of the quadratic formula.
If b^2 - 4ac is a positive perfect square, then it can be factored.
x^2 + 100 ---> 1*x^2 + 0*x + 100 ---> a=1, b=0, c=100
b^2 - 4ac = (0)^2 - 4(1)(100) = 0^2 - 400 = 0 - 400 = -400
It is negative, meaning the zeros are complex/imaginary.
So no it cannot be factored.
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Now let's change the problem just a little bit....
x^2 - 100 = x^2 + 0x - 100
(x + )(x - ) <---- again, the only way to get x^2 is x times x;
the constant term is negative, so the signs are different
Now we need two numbers that multiply to -100 and add up to zero.
If you look in the list shown in the previous problem, it is 10 and -10.
So (x+10)(x-10)
Please verify by FOILing.
This is called the DIFFERENCE of square pattern.
X^2 - C^2 for some number C is
(X+C)(X-C)
That is x^2 minus a perfect square will factor as X plus the square root of the constant times
X minus the square root of the constant
Practice these, please. Some cannot be factored. If so, factor it. If not, say NO!!!
x^2 - 16
x^2 - 49
x^2 + 36
x^2 - 64
x^2 - 25
x^2 + 81
Can you factor 16 - x^2
Please practice these, and tell us if you still do not understand.
Thanks!!!
