Choose one of the factors of x6 + 1000.
x2, x2 - 10, x4 - 10x2 + 100, x4 + 10x2 + 100
Solution:
Let us use (a + b)3 identity to factorize the expression. (a + b)3 = a3 + b3 + 3ab2 + 3a2b can be rewritten as
⇒ a3 + b3 = (a + b)3 - 3ab2 - 3a2b --- (1)
Therefore we can write the given expression x6 + 1000 as per equation(1)
x6 + 1000 = (x2)3 + (10)3 = (x2 + 10)3 - 3(x2)(10)2 - 3(x2)2(10) --- (2)
Where x2 corresponds to “a” in equation(1) and 10 corresponds to “b” in equation(1)
Rewriting equation(2) we have,
(x2 + 10)3 - 300x2 - 30x4
⇒ (x2 + 10)3 - 30x2(x2 + 10) --- (3)
Taking (x2 + 10) as the common factor we have
(x2 + 10){(x2 + 10)2 - 30x2}
⇒ (x2 + 10)(x4 + 100 + 20x2 - 30x2)
⇒ (x2 + 10)(x4 + 100 + 20x2 - 30x2)
⇒ (x2 + 10)(x4 - 10x2 + 100 )
The above expressions show x6 + 1000 has two factors and they are
(x2 + 10) & (x4 - 10x2 + 100) and one of the two factors is (x4 - 10x2 + 100) which is the correct alternative of the choices given in the problem.
Choose one of the factors of x6 + 1000.
Summary:
One of the other factor of x6 + 1000 is x4 - 10x2 + 100.
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