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) = x^4 - 4x^3 - 15x^2 + 58x - 40 $ $ (x - 5) $, $ (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) = x^4 - 4x^3 - 15x^2 + 58x - 40 $ $ (x - 5) $, $ (x + 4) $

Transcript

00:01 We're going to start by verifying the given factors.

00:04 So i'm going to start with x minus 5 and use synthetic division to show that it does divide evenly into f of x.

00:12 So i'm going to put five outside the box and then the coefficients of the polynomial, 1x to the 4, negative 4x cubed, negative 15x squared, 58x, and negative 40.

00:24 So we bring down the 1, multiply it by 5, we get 5, write it in the next space and add the column, and we have 1.

00:32 That by 5, we have 5, write that in the next space, add the column, and we get negative 10.

00:37 Multiply that by 5, we get negative 50, write that in the next space, and add the column, and we get 8.

00:43 Multiply that by 5, we get 40, write that in the next space, and add the column, and we get 0.

00:48 So that confirms that we do have a factor, and what we can do next is take what we just got and divide the other factor x plus 4 into it by putting negative 4 outside the box, 1 ,1, negative 10, 8 are the coefficients of what remains.

01:09 Bring down the 1.

01:10 Multiply by negative 4, we get negative 4.

01:13 Write that in the next space and add the column.

01:14 We get negative 3.

01:16 Multiply that by negative 4, we get 12.

01:18 Put that in the next space and add the column, and we get 2.

01:21 Multiply that by negative 4, we get negative 8.

01:24 Put that in the next space and add the column when we get 0.

01:26 So that confirms that x plus 4 is also a factor.

01:31 So that takes care of part a.

01:33 Now we're on to part b where we want to find all the factors.

01:37 So we already have x minus 5 and we have x plus 4.