B How can you factorize polynomials with fourth degree terms?

[gede]

Since this is not available in my algebra textbook, how do you factorize the $a^4 + b^4$ and $a^4 - b^4$?

I also would like to know how do you obtain $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$?

 

[SteamKing]

Since this is not available in my algebra textbook, how do you factorize the $a^4 + b^4$ and $a^4 - b^4$?

I also would like to know how do you obtain $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ and $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$?

Factoring a⁴ + b⁴ is tricky.

Factoring a⁴ - b⁴ should be obvious if you know how to factor a² - b²

For the factoring of the cubics (or polynomials in general), you can always use polynomial long division.

 

[Mark44]

Factoring a⁴ + b⁴ is tricky.

And if you're limited to factors with real coefficients, it's not factorable at all.

 

[Citan Uzuki]

And if you're limited to factors with real coefficients, it's not factorable at all.

Actually it is. It factors as (a^2 + \sqrt{2}ab + b^2)(a^2 - \sqrt{2}ab + b^2)

 

[member 587159]

1) a^4 - b^4 = ...

Substituting a^2 = x and b^2 = y

=> x^2 - y^2 = ...

2) a^4 + b^4 = a^4 + 2a^2*b^2 + b^4 - 2a^2*b^2 = (a^2 + b^2)^2 - 2a^2*b^2

Now use a^2 - b^2 = (a-b)(a+b) to find the answer

3) a^3 - b^3 = (a-b)(a^2 + ab + b^2)
=> (a^3 - b^3 )/(a-b) = (a^2 + ab + b^2)

Use euclidean division or Horner (let a^3 be the variable, let b^3 be the constant)

 

[Noriele Cruz]

There's a trick in factoring (a³ - b³) and any expression alike of this. (Only works for this kind of binomial).

1) The first factor, which is a binomial, is always the cube root of the two terms.
(The blank space represents the missing terms and its operation/ sign).

a³ - b³ = (a - b)( _ _ _ )

2) The second factor, which is a trinomial, always constitutes three terms as follows.
a. The first term is square of the first term in the binomial: (a - b)(a² _ _ )
b. The middle term is reverse sign of the product of the two term: (a - b)(a² + ab _)
c. The last term is the square of the second term in the binomial: (a - b)(a² + ab + b²)

And that completes the factoring of (a³ - b³)

 

[member 587159]

There's a trick in factoring (a³ - b³) and any expression alike of this. (Only works for this kind of binomial).

1) The first factor, which is a binomial, is always the cube root of the two terms.
(The blank space represents the missing terms and its operation/ sign).

a³ - b³ = (a - b)( _ _ _ )

2) The second factor, which is a trinomial, always constitutes three terms as follows.
a. The first term is square of the first term in the binomial: (a - b)(a² _ _ )
b. The middle term is reverse sign of the product of the two term: (a - b)(a² + ab _)
c. The last term is the square of the second term in the binomial: (a - b)(a² + ab + b²)

And that completes the factoring of (a³ - b³)

This is true, although the point of mathematics is not to remember some tricks. With some experience, it takes 30 seconds to proof this formula using general calculus.

 

[Noriele Cruz]

This is true, although the point of mathematics is not to remember some tricks. With some experience, it takes 30 seconds to proof this formula using general calculus.

Nonetheless, it is a basic format on how to factor such kind of expressions. It's not a point of remembering a trick, it is one way of knowing a proof how such expression can be branched down into its respective factors.

 

[Mark44]

This is true, although the point of mathematics is not to remember some tricks.

It's a huge time saver to remember a few "tricks", such as the Quadratic Formula and how to factor the difference of squares and the sum or difference of cubes.

With some experience, it takes 30 seconds to proof this formula using general calculus.

You don't need calculus to derive $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Also, since the OP is studying algebra, it's reasonable to assume that he hasn't been exposed to the techniques of calculus.

 

[micromass]

This is true, although the point of mathematics is not to remember some tricks. With some experience, it takes 30 seconds to proof this formula using general calculus.

How would you prove this using calculus?

 

[gede]

Please tell me the method of factoring of $a^n + b^n$ and $a^n - b^n$?

 

[member 587159]

Please tell me the method of factoring of $a^n + b^n$ and $a^n - b^n$?

1) Define a function F: R -> R: a -> a^n + b^n => F(a) = a^n + b^n
b and n are natural numbers, n is odd

We see: F(-b) = -b^n + b^n = 0 => a + b is a divisor of F(a) (fundamental theorem of algebra)
That's one factor, you can find what's left after the division using horner's rule or using euclidean division. Then you can try to find those factors.

Same thing when you have Z(a) = a^n -b^n
b and n natural numbers, n odd

2) Define a function G: R -> R: a -> a^n - b^n => G(a) = a^n - b^n
b and n are natural numbers, n is not odd

Use a^2 - b^2 = (a-b)(a+b) to find factors.

3) Define a function H: R -> R: a -> a^n + b^n => H(a) = a^n + b^n
b and n are natural numbers, n is not odd

This is the hardest one. You have to manipulate the expressions. For example,
a^4 +1 = a^4 + 1 + 2a^2 - 2a^2 = (a^2 +1)^2 - 2a^2 = (a^2 + 1 + SQRT(2)a)(a^2 + 1 - SQRT(2)a)

 

[Mark44]

1) Define a function F: R -> R: a -> a^n + b^n => F(a) = a^n + b^n
b and n are natural numbers, n is odd

We see: F(-b) = -b^n + b^n = 0 => a + b is a divisor of F(a) (fundamental theorem of algebra)
That's one factor, you can find what's left after the division using horner's rule or using euclidean division. Then you can try to find those factors.

Same thing when you have Z(a) = a^n -b^n
b and n natural numbers, n odd

2) Define a function G: R -> R: a -> a^n - b^n => G(a) = a^n - b^n
b and n are natural numbers, n is not odd

Use a^2 - b^2 = (a-b)(a+b) to find factors.

3) Define a function H: R -> R: a -> a^n + b^n => H(a) = a^n + b^n
b and n are natural numbers, n is not odd

This is the hardest one. You have to manipulate the expressions. For example,
a^4 +1 = a^4 + 1 + 2a^2 - 2a^2 = (a^2 +1)^2 - 2a^2 = (a^2 + 1 + SQRT(2)a)(a^2 + 1 - SQRT(2)a)

None of these techniques use calculus, which is what @micromass asked about. For $a^4 + 1$, if by factoring, one means splitting the polynomial into linear factors (i.e., first degree factors) with real coefficients, it's not factorable.

That was what I was thinking when I said that $a^4 + b^4$ wasn't factorable. To clarify my statement, $a^4 + b^4$ isn't factorable into linear factors with real coefficients.