Answer:
x = -4, -1, 9
Explanation:
The zeros of f(x) will be found from the factors of f(x). Like factoring numbers, you reduce the problem in stages by factoring out the given factor. You can do this using polynomial long division, or its simpler cousin, synthetic division.
The attachment shows the synthetic division required to determine that the remaining quadratic factor is (x^2 -8x -9). That is ...
f(x) = (x +4)(x^2 -8x -9)
The quadratic can be factored by looking for two numbers that have a product of -9, but have a sum of -8. Those numbers are +1 and -9. These are the values of the constants in the remaining factors you need.
f(x) = (x +4)(x +1)(x -9)
The zeros of f(x) are the values of x that make these factors zero. (When a factor is 0, f(x) is 0.) The values of interest are the opposites of the constant in each factor, -4, -1, 9.
The zeros of f(x) are x = -4, -1, 9.
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Additional comment
In regular division, the divisor is multiplied by the (partial) quotient, and the product is subtracted from the dividend to get a new dividend. In synthetic division, the divisor is presumed to be a binomial of the form (x -r), where r is the zero of this binomial term. (x=r gives r-r = 0) It is this r value that appears in the upper left corner of the synthetic division table.