Find the value of k if (x-2) is a factor of x³+2x²-kx+10. Determine whether (x+5) is also a factor.
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Solution!!
The concept of Factor Theorem and Remainder Theorem has to be used here.
Factor Theorem → When a polynomial f(x) is divided by x - a, the remainder is f(a). And, if the remainder f(a) is 0; x - a is a factor of the polynomial f(x).
Remainder Theorem → If f(x), a polynomial in x, is divided by (x - a), the remainder is f(a).
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x - 2 is a factor.
x - 2 = 0
x = 2
f(x) = x³ + 2x² - kx + 10
f(2) = (2)³ + 2(2)² - k(2) + 10
0 = 8 + 8 - 2k + 10
0 = 26 - 2k
2k = 26
k = 13
Now let's see if (x + 5) is a factor of the expression or not.
x + 5 = 0
x = -5
f(x) = x³ + 2x² - kx + 10
f(-5) = (-5)³ + 2(-5)² - 13(-5) + 10
= -125 + 50 + 65 + 10
= -125 + 125
= 0
Hence, (x + 5) is also a factor.
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