The function f(x) = 3x^3 + 14x^2 + 13x

The function $f(x) = 3x^3 + 14x^2 + 13x - 6$ has a zero at $x = -3$. a. What factor must be included in the factored form for $f$ based on this information? (Hint: Your answer will be something like $x - 1$ or $x + 1$.) b. Use synthetic division to factor out the expression you wrote in part (a), then use the result to write the fully factored form of $f$ below. Your final answer should not have any factors with $x^2$ in them. $f(x) =$ c. Write the function zeros below separated by commas. $x =$ Box 1: Enter your answer as an expression. Example: $3x^2+1$, $x/5$, $(a+b)/c$

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00:01 This problem gives us a function, and it wants us to do a few things, given a few factors, and first is to verify the given factors of f of x by completing the synthetic division that we see, and then using our results to find the remaining factors of f of x, and writing the complete factorization, and then the zeros of the function.

00:16 So to finish the synthetic division to prove these are both factors, we will pick up with following their steps, and then fill in the blanks as we go.

00:23 They drop down the 2 to get 2, and then 2 times negative 3 gives us negative 6, 1 plus negative 6 gives us negative 5.

00:30 Negative 5 times negative 3 gives us a positive 15.

00:33 Negative 13 plus 15 gives us 2.

00:36 2 times negative 3 gives us negative 6.

00:38 6 plus negative 6 gives us 0.

00:40 So since the remainder is 0, that proves that x plus 3 was a factor.

00:45 For x minus 2, same thing.

00:47 We bring down the 2 first.

00:48 2 times 2 gives us 4.

00:50 1 plus 4 gives us 5.

00:51 5 times the 2 gives us 10.

00:53 Negative 13 plus 10 gives us negative 3.

00:56 Negative 3 times 2 gives us negative 6.

00:58 And once again, 6 plus negative 6 gives us 0.

01:01 So the remainder 0, that means this is a factor.

01:04 What we can do now to be able to factor is to, we could look at the results of factoring our two factors out...

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