Math Is Fun Forum

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Factor Completely

Factor completely.  If the polynomial cannot be factored, say it is prime.

x^6 + 2x^3 + 1

Bob

Administrator

Registered: 2010-06-20

Posts: 10,631

Re: Factor Completely

We're in luck here because (x^3)^2 = x^6

So substitute Y = x^3 and you'll get a quadratic in Y.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Factor Completely

Bob,

Thank you. You said we're in luck in this case because (x^3)^2 = x^6.
What if the question is not so obvious? What then?

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Factor Completely

Factor completely.  If the polynomial cannot be factored, say it is prime.

x^6 + 2x^3 + 1

We're in luck here because (x^3)^2 = x^6

You said we're in luck in this case because (x^3)^2 = x^6.
What if the question is not so obvious? What then?

then maybe it cant be factored, right?
if your doing perfect sqares then it has to fit into perfect aquare pattern

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Factor Completely

We're in luck here because (x^3)^2 = x^6

So substitute Y = x^3 and you'll get a quadratic in Y.

Bob

Let me see.

Rewrite x^6 as (x^3)^2.

Let u = x^3

u^2 + 2u + 1

Factor.

(u + 1)(u + 1)

Back-substitute for u.

(x^3 + 1)(x^3 +1)

The sum of cubes tells me that (x^3 + 1) is the same thing as
(x + 1)(x^2 - x  + 1).

My answer is (x + 1)(x^2 - x + 1)^2.

Textbook answer is (x + 1)^2(x^2 - x + 1)^2.

Who is right and why?

amnkb

Member

Registered: 2023-09-19

Posts: 253

Re: Factor Completely

Rewrite x^6 as (x^3)^2.

Let u = x^3

u^2 + 2u + 1

Factor.

(u + 1)(u + 1)

Back-substitute for u.

(x^3 + 1)(x^3 +1)

The sum of cubes tells me that (x^3 + 1) is the same thing as
(x + 1)(x^2 - x  + 1).

My answer is (x + 1)(x^2 - x + 1)^2.

Textbook answer is (x + 1)^2(x^2 - x + 1)^2.

Who is right and why?

in the red part you have

in the blue part you factored

then replace

the aquare goes on both factors inside
once you do that youll match the books answer

sologuitar

Member

Registered: 2022-09-19

Posts: 467

Re: Factor Completely

Thank you.