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) = 2x^3 + x^2 - 5x + 2 $ $ (x + 2) $, $ (x - 1) $

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) = 2x^3 + x^2 - 5x + 2 $ $ (x + 2) $, $ (x - 1) $

Transcript

00:01 Okay, let's start this problem by verifying that x plus 2 and x minus 1 are factors by using synthetic division.

00:10 So for x plus 2, i'm going to put a negative 2 outside the box, and then the coefficients of the polynomial, 2, 1, negative 5, 2.

00:19 And then we're looking to see that we get a remainder of 0.

00:22 Bring down the 2, multiply it by negative 2, write it in the next space, and add the column.

00:27 Multiply this by negative 2, write it in the next space and add the column.

00:31 Multiply this by negative 2, write it in the next space and add the column.

00:35 So yes, we do get a remainder of 0, so x plus 2 is a factor.

00:40 Let's do the same for x minus 1.

00:41 So we put 1 outside the box.

00:44 We have our same coefficients.

00:47 Bring down the first number 2.

00:49 Multiply it by 1, write it in the next space and add the column.

00:53 Multiply 3 by 1, write it in the next space and add the column.

00:57 Multiplied negative 2 by 1, read it in the next space and add the column.

01:00 So once again, we get a remainder of 0, so we have verified the factors.

01:04 So that's part a.

01:09 Part b, we want to find the additional factors.

01:12 So what i'm going to do is go back to part a, go back to the first synthetic division, look at those remaining coefficients, and realize that those are the coefficients on the quadratic, 2x squared minus 3x plus 1.

01:25 So that multiplied by x plus 2 is our polynomial...

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