Answer:
Explanation:
You want the zeros of f(x) = x³ -5x² -22x -16, given that a factor is x+2.
Factors
Using synthetic division (see attachment), we find the quadratic factor to be (x² -7x -8), so we have ...
f(x) = (x +2)(x² -7x -8)
The quadratic can be factored using our knowledge of the divisors of -8 to give ...
f(x) = (x +2)(x +1)(x -8)
Zeros
The zeros of f(x) are the values of x that make these factors zero:
x = -2, -1, 8 . . . . . zeros of f(x)
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Additional comment
We know the product of binomial factors is ...
(x -a)(x +b) = x² -(a-b)x -ab
This means we can factor the quadratic by looking for factors of 8 that have a difference of 7. We know that 8 = 8·1 and that 8-1=7, so the values of 'a' and 'b' we're looking for are a=8, b=1.
The "zero product rule" tells you a product is zero only if one of the factors is zero. That is how we know to look for the zeros of the binomial factors of f(x). For example, x+2=0 ⇒ x=-2 is a zero of f(x). (The remainder of 0 in the synthetic division also tells us f(-2)=0.)
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