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Question
Show that x2 - 9 is factor of x3 + 5x2 - 9x - 45.
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Solution
We know that
x2 - 9 = (x + 3)(x - 3)
x2 - 9 will be a factor of
f(x) = x3 + 5x2 - 9x - 45
Only when both x + 3 are factor of this polynomial.
Now, f(-3) = (-3)3 + 5(-3)2 -9(-3) -45
= -27 + 45 + 27 - 45 = 0
And f(3) = (3)3 + 5(3)2 - 9(3) -45
= 27 + 45 - 27 - 45 = 0
So, both x + 3 and x - 3 are factor of x3 + 5x2 - 9x - 45.
Hence, x2 - 9 is a factor of the given polynomial.
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