Question

Factorise the following equation
x⁸+1

Answer 1

Step-by-step explanation:

Step by Step Solution

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Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x8" was replaced by "x^8".

STEP

1

:

Trying to factor as a Difference of Squares

1.1 Factoring: x8-1

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1

Check : x8 is the square of x4

Factorization is : (x4 + 1) • (x4 - 1)

Polynomial Roots Calculator :

1.2 Find roots (zeroes) of : F(x) = x4 + 1

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 1.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 2.00

1 1 1.00 2.00

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares:

1.3 Factoring: x4 - 1

Check : 1 is the square of 1

Check : x4 is the square of x2

Factorization is : (x2 + 1) • (x2 - 1)

Polynomial Roots Calculator :

1.4 Find roots (zeroes) of : F(x) = x2 + 1

See theory in step 1.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is 1.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 2.00

1 1 1.00 2.00

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares:

1.5 Factoring: x2 - 1

Check : 1 is the square of 1

Check : x2 is the square of x1

Factorization is : (x + 1) • (x - 1)

Final result :

(x4 + 1) • (x2 + 1) • (x + 1) • (x-1)

Answer 2

Answer:

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Step-by-step explanation:

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1 is the square of 1

Check : x8 is the square of x4

Factorization is : (x4 + 1) • (x4 - 1)