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To divide the function f(x) by (x^2-4x-12), we find that the quotient is Q(x)
and the remainder is (x+6). If f(-2)=a and f(6)=b, what is the value of b/a?
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First of all, the problem is worded incorrectly. To be correct, it must sound THIS WAY:
To divide the POLYNOMIAL f(x) by (x^2-4x-12), we find that the quotient is Q(x)
and the remainder is (x+6). If f(-2)=a and f(6)=b, what is the value of b/a?
With this correction, see my solution below.
From the condition, we have this polynomial equality
f(x) = (x^2-4x-12)*Q(x) + (x+6), (1)
where Q(x) is some polynomial (the quotient of the division).
Notice that the polynomial (x^2 -4x - 12) has the roots -2 and 6:
exactly the values mentioned in the problem's condition.
THEREFORE, substituting the values x= -2 and x= 6 into formula (1), we get
a = f(-2) = 0*Q(-2) + (-2+6) = 0 + 4 = 4,
b = f(6) = 0*Q(6) + (6+6) = 0 + 12 = 12.
Hence, = = 3.
ANSWER. The value of is 3.
Solved.