Factoring A Polynomial...1

[nycmathguy]

Section 2.3
Question 64

Can you do 64 parts (a - e) as a guide for me to try a few more on my own?

 

[MathLover1]

f(x)=3x^3-x^2-8x-4

factors:
(x+1), (x-2)

a. verify the given factors

use synthetic division: if (x+1)=0=>x=-1

-1| 3 -1 -8 -4
|..........-3 4 4
|___________________
....3 -4 -4 0 =>resulting coefficients: 3,-4,-4,0

the quotient is 3x^2-4x-4, and the remainder is 0


(x-2)=0=>x=2
2| 3 -1 -8 -4
|..........6 10 4
|___________________
....3 5 2 0 =>resulting coefficients: 3,5,2,0

the quotient is 3x^2+5x+2, and the remainder is 0.


b. find the remaining factor(s)

use long division
3x^3-x^2-8x-4 divide by (x+1)(x-2)=x^2 - x - 2

...................(3x+2
(x^2 - x - 2)|3x^3-x^2-8x-4
....................3x^3-3x^2-6x
..............................2x^2-2x-4
..............................2x^2-2x-4
.............................................0

the remaining factor is (3x+2)

c. use your results to write complete factorization

(3 x + 2) (x + 1) (x - 2)


d. list all zeros
(3 x + 2) =0=>3x=-2=>x=-2/3
(x + 1) =0=>x=-1
(x - 2)=0=>x=2

e. confirm your results by using graph

 

[nycmathguy]

f(x)=3x^3-x^2-8x-4

factors:
(x+1), (x-2)

a. verify the given factors

use synthetic division: if (x+1)=0=>x=-1

-1| 3 -1 -8 -4
|..........-3 4 4
|___________________
....3 -4 -4 0 =>resulting coefficients: 3,-4,-4,0

the quotient is 3x^2-4x-4, and the remainder is 0


(x-2)=0=>x=2
2| 3 -1 -8 -4
|..........6 10 4
|___________________
....3 5 2 0 =>resulting coefficients: 3,5,2,0

the quotient is 3x^2+5x+2, and the remainder is 0.


b. find the remaining factor(s)

use long division
3x^3-x^2-8x-4 divide by (x+1)(x-2)=x^2 - x - 2

...................(3x+2
(x^2 - x - 2)|3x^3-x^2-8x-4
....................3x^3-3x^2-6x
..............................2x^2-2x-4
..............................2x^2-2x-4
.............................................0

the remaining factor is (3x+2)

c. use your results to write complete factorization

(3 x + 2) (x + 1) (x - 2)


d. list all zeros
(3 x + 2) =0=>3x=-2=>x=-2/3
(x + 1) =0=>x=-1
(x - 2)=0=>x=2

e. confirm your results by using graph


View attachment 402

I don't recall how to use long division when the divisor is a trinomial. Can you post a video clip showing how this is done?

 

[MathLover1]

the process is same as when the divisor is a binomial

 

[nycmathguy]

the process is same as when the divisor is a binomial

This video clip was very helpful. I don't care too much for the teacher. He makes too many mistakes. I have seen him on numerous occasions. The students constantly correcting his work. I wonder how he became a math teacher.

 

[nycmathguy]

f(x)=3x^3-x^2-8x-4

factors:
(x+1), (x-2)

a. verify the given factors

use synthetic division: if (x+1)=0=>x=-1

-1| 3 -1 -8 -4
|..........-3 4 4
|___________________
....3 -4 -4 0 =>resulting coefficients: 3,-4,-4,0

the quotient is 3x^2-4x-4, and the remainder is 0


(x-2)=0=>x=2
2| 3 -1 -8 -4
|..........6 10 4
|___________________
....3 5 2 0 =>resulting coefficients: 3,5,2,0

the quotient is 3x^2+5x+2, and the remainder is 0.


b. find the remaining factor(s)

use long division
3x^3-x^2-8x-4 divide by (x+1)(x-2)=x^2 - x - 2

...................(3x+2
(x^2 - x - 2)|3x^3-x^2-8x-4
....................3x^3-3x^2-6x
..............................2x^2-2x-4
..............................2x^2-2x-4
.............................................0

the remaining factor is (3x+2)

c. use your results to write complete factorization

(3 x + 2) (x + 1) (x - 2)


d. list all zeros
(3 x + 2) =0=>3x=-2=>x=-2/3
(x + 1) =0=>x=-1
(x - 2)=0=>x=2

e. confirm your results by using graph


View attachment 402

Perfect. I can use this reply to answer a few more if I decide to do so. Moving on.