In Exercises 67 - 74, (a) verify the given factors of the...

In Exercises 67 - 74, (a) verify the given factors of the function $ f $,(b) find the remaining factor(s) of $ f $ (c) use your results to write the complete factorization of $ f $, (d) list all real zeros of $ f $, and (e) confirm your results by using a graphing utility to graph the function. Function Factors $ f(x) = 8x^4 - 14x^3 - 71x^2 - 10x + 24 $ $ (x + 2) $, $ (x - 4) $

In Exercises 67 - 74, (a) verify the given factors of the function $ f $,(b) find the remaining factor(s) of $ f $ (c) use your results to write the complete factorization of $ f $, (d) list all real zeros of $ f $, and (e) confirm your results by using a graphing utility to graph the function.

Function Factors
$ f(x) = 8x^4 - 14x^3 - 71x^2 - 10x + 24 $ $ (x + 2) $, $ (x - 4) $

Transcript

00:01 We're going to start by using synthetic division to show that x plus 2 is a factor of f of x.

00:07 So i'm going to put negative 2 outside the box, and then the coefficients of the polynomial go in a row.

00:13 So we have 8x to the 4, negative 14x cubed, negative 71x squared, negative 10x, and 24.

00:22 We bring down the 8.

00:24 Multiply it by negative 2, we get negative 16.

00:27 Write it down and add the column, and we get negative 30.

00:30 Multiply that by negative 2, we get 60, write it down and add the column, and we get negative 11.

00:36 Multiply that by negative 2, and we get 22, write it down and add the column, and we get 12.

00:42 Multiply that by negative 2, we get negative 24, write it down and add the column, and we get 0.

00:47 So that means the remainder is 0, so x plus 2 is a factor, since that went in evenly.

00:53 We also want to show that x minus 4 is a factor, so what we can do is divide it into the polynomial we have left, now that we have already divided by x plus 2.

01:03 So that means we're going to use these numbers.

01:05 So we're going to put four outside the box.

01:08 We're going to use 8, negative 30, negative 11, and 12.

01:13 Bring down the 8.

01:15 Multiply that by 4, we get 32.

01:17 Write it down and add the column, and we have 2.

01:20 Multiply that by 4 and we have 8.

01:22 Write it down and add the column, and we get negative 3.

01:25 Multiply that by 4 and we have negative 12.

01:27 Write it down and add the column, and we get 0.

01:29 So again, a remainder of zero indicates that we have a factor.

01:33 It went in evenly.

01:34 So at this point, we've finished part a, we verified those two factors, and we're going to start part b finding all factors.

01:42 So we already have the factor x plus two, and we have the factor x minus four.

01:47 What we're going to do to find the remaining two factors is take the numbers that were left after our last synthetic division and realize that those represent the remaining quadratic, 8x squared plus 2x minus 3...

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