Question

I don't want a direct answer, I want someone to help me understand and walk me through the question.

Question:

Bo factored x^2 – 10xy + 16y^2 as follows:


Line 1 x2 – 10xy + 16y2

Line 2 x2 – 8xy – 2xy + 16y2

Line 3 x(x – 8) – 2y(x – 8y)

Line 4 prime


A. Did Bo correctly factor the polynomial? Justify your response by explaining why it is is factored correctly or by showing the correct solution if it is not factored correctly.

B. Explain how you would know if a polynomial of the form ax^2 + bxy + cy^2 was prime.

Someone help plz, I'll give medals.

Answer 1

@triciaal @zepdrix @mathmale @mathstudent55 @SolomonZelman

Answer 2

idk srry

Answer 3

@tygrr321 Alright, that's fine. Thanks anyways.

Answer 4

@ParthKohli @imqwerty @Directrix @robtobey @SolomonZelman

Answer 5

When x2 – 8xy was factored, the Bo got x ( x - 8).

What happened to the y in the term -8xy ?

What should Bo get?

Answer 6

@Directrix Bo should have gotten x(x - 8y)?

Answer 7

x2 – 8xy = x ( x - 8y) That is correct. @Needhelpstudying

Answer 8

Hooray!

Answer 9

a) Okay... the correct way is

\[x(x – 8y) – 2y(x – 8y)\]

Due to this bo did not factor correctly.

So, the best solution would be:

\[x^2 – 10xy + 16y^2 = (x - 2y) (x - 8y)\]

b) If both a and c are prime then the polynomial is prime. For example, \[X^2+5\] is prime.

Do you understand?

But, overall I think @Directrix will help completely.

I just wanted to add in my thoughts :)

But, @Directrix will explain better :)

Answer 10

Hope this helped! Have a great day and Happy Holidays! :)

Also, a fan will be much appreciated as well! Just hover over my avatar and click “Become a fan”. This will tell you whenever I am online so you can ask me for help in any of your other questions!

If you see that I am online and need help with a question, just tag me in your question!

@Needhelpstudying

Answer 11

@Needhelpstudying

Start at line 2 and work the problem for Bo.

Line 2 x2 – 8xy – 2xy + 16y2

Answer 12

@Directrix Alright

Answer 13

x2 - 8xy - 2xy + 16y2 = (x2 - 8xy) - (2xy + 16y2)

Answer 14

Find GCF:

GCF for first binomial: x

GCF for second binomial: 2y

Factor out GCF: x(x - 8y) - 2y(x - 8y)

Answer 15

Combine binomials:

(x - 2y)(x - 8y) = FINAL Answer

Answer 16

Line 2 x2 – 8xy – 2xy + 16y2

Line 3: x( x - 8y) - 2 y ( x - 8y)

(x - 8y) is a common factor

Line 4: ( x - 8y) ( x - 2y)

Answer 17

Correct.

Answer 18

What about this:

B. Explain how you would know if a polynomial of the form ax^2 + bxy + cy^2 was prime.

Answer 19

Alright, so now I know that Bo didn't factor the polynomial correctly, and I know what the correct answer is. Part A is done

Answer 20

@Directrix What exactly does a prime polynomial mean?

Answer 21

Yes and part B is up to solve.

Answer 22

A prime number such as 3 will not factor (other than 1*3).

A prime polynomial will not factor.

Answer 23

@Directrix Sorry, my laptop shut off

Answer 24

@Directrix B. Explain how you would know if a polynomial of the form ax^2 + bxy + cy^2 was prime.

I would know if a polynomial of the form ax^2 + bxy + cy^2 was prime, because it would not factor at all/any further.

Would that answer be correct?

Answer 25

@Directrix Hello??

Answer 26

@Directrix Was I correct?

Answer 27

>I would know if a polynomial of the form ax^2 + bxy + cy^2 was prime, because it would not factor at all/any further.

While that is true, I think that the question is asking for a more detailed answer.

Answer 28

@Directrix Like give an example of a prime polynomial and show that it's prime?

Answer 29

@AihberKhan posted this above:

>>b) If both a and c are prime then the polynomial is prime.

I saw that over at Yahoo answers but there was no explanation so I do not know if it is true.

Answer 30

@Directrix Any way you can find out if it's true?

Answer 31

We could test it by seeing what happens when factoring a polynomial where a and c are prime.

3x^2 + 5x - 2

3 is prime and 2 is prime.

Can you factor: 3x^2 + 5x - 2 @Needhelpstudying

Answer 32

Yea sure

Answer 33

3x^2 + 5x - 2 = 3x^2 + 3x + 2x - 2

Group them together:

(3x^2 + 3x) + (2x - 2)

Find GCF

GCF of first binomial is 3x

GCF of second binomial is 2

Divide GCF

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

Answer 34

Add terms together:

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

Wait... wut?

Answer 35

@Directrix help

Answer 36

Did an online calculator do the factoring you posted? :)

Answer 37

No, but I did have to use a calculator to find out what -2 / 2 was. (My brain turned off)

Answer 38

3x^2 + 5x - 2

Find two numbers that multiply to a*c and add to b

a = 3
c = -2
b = 5

Find two numbers that multiply to -6 and add to 5 @Needhelpstudying

Answer 39

Why?

Answer 40

>>Why? That is the procedure for factoring by grouping.'


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

3 and 2 are prime so I don't know what to say about @AihberKhan 's post that

>>"b) If both a and c are prime then the polynomial is prime."

It seems not to be true in this case.

Answer 41

Do you think my teacher will mind if all I say is something like

"I would know if a polynomial of the form ax^2 + bxy + cy^2 was prime, because it would not factor at all/any further. "

Answer 42

@Directrix Cause technically thats what theyre asking

Answer 43

I see your point. I don't know for sure what the teacher will accept.

Go with it and see.

Answer 44

Alright. Thanks for all your help on this @Directrix !!!!

Answer 45

@AihberKhan

Would you explain this. Thanks.

> b) If both a and c are prime then the polynomial is prime.

Answer 46

brb

Answer 47

Okay. Let me explain part b "_

So basically if a in the equation and c in the equation are prime... this simply means that the whole equation is prime..

As I said in my answer... for example...

\[x^2 + 5\]

is prime.

So if a and c look like this it is prime..

You can then conclude that the entire equation is prime..

Hope this helped further explain! :)

Sorry of this doesn't helped I don't know how I can explain it any further but good luck anyways!!

@Needhelpstudying

Have a great day and Happy Holidays! :)

Answer 48

@AihberKhan

What about this where a and c are prime?

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

3 and 2 are prime so I don't know what to say about @AihberKhan 's post that

>>"b) If both a and c are prime then the polynomial is prime."

It seems not to be true in this case.

Answer 49

@Directrix While you guys sort this out, ima write my answer for part a.

Answer 50

Oh I see.

I am truly sorry, I did not mean to give false information by any means! Thank you so much @Directrix for correcting me!

I truly apologize for giving any false information... just wanted to add in a little thought.

Thank you @Directrix for correcting me! I will be sure to think about that in the future! :)

Answer 51

:) ima give the proof why the polynomial will be considered prime is \(a\) and \(c\) were prime
lets say the polynomial is- \(ac^2+bx+c\)
we know that the product of roots is given by->\(\large\frac{c}{a}\)
lets say that the roots are-\(\alpha\) and \(\beta\)
if both \(c\) and \(a\) are prime then -
\(\large \alpha \times \beta =\frac{c}{a}\)
now because a and c are prime either of \(\alpha\) or \(\beta\) or both will have to be in fraction form
and we know that the polynomial \(ax^2+bx+c\) can be when written in factored form is this- \((x-\alpha)(x-\beta)\)
since the roots are in fraction form the factors are not integers and thus the polynomial has no integer divisors so it will be a prime polynomial

Answer 52

No need to apologize. @AihberKhan We are all just thinking about this problem.

Answer 53

Oh okay great! :)

@imqwerty

Answer 54

:) okay now for part A
he did the mistake in 2nd step

he used this method-> splitting the middle term and he did it wrong

reason why he did it wrong->
lets say the polynomial is -> \(ax^2+bx+c\)
when we split the middle term \(b\) we split it in such a way that the product of the 2 things into which we split our middle term must be equal to this product- \(ax^2 \times c\)
and the sum of the two things into which we split must be equal to \(bx\)

Answer 55

Alright guys, thanks for all your help on this

Answer 56

for part B-> the polynomial is prime when the factors are not integral :)

Answer 57

and we know what is the situation when this occurs :)
when a and c both become prime