Answer :
Final answer:
Given that 5 is a zero of the function f(x)=x³-5x²+9x-45, we can say (x-5) is a factor. We get the other factors by dividing f(x) by (x-5), yielding (x²+2) as the remaining factor. The roots are x=5 and x=±√-2.
Explanation:
To factor the function f(x)=x³-5x²+9x-45 into factors of the form (x-c), we know one of the factors because the problem tells us 5 is a zero of the function. Therefore, (x-5) is a factor of the polynomial.
We find the other factors by performing polynomial division or synthetic division of f(x) by (x-5). The quotient we get from this division is the other factor of f(x).
For example, by synthetic division with 5, we get a quotient of x2+2, therefore the factorization of the function is (x-5)(x²+2).
The roots or zeros of the function can be found by setting each factor equal to 0 and solving for x. For (x-5), that gives us one root at x=5. For (x²+2), setting it equal to zero gives us x²=-2, or x=±√-2, which are complex roots.
Learn more about factoring polynomials here:
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