Question

what is the factor of 4x⁴+1​

Answer 1

Step-by-step explanation:

(1): "x4" was replaced by "x^4".

STEP

1

:

Equation at the end of step 1

22x4 - 1

STEP

2

:

Trying to factor as a Difference of Squares

2.1 Factoring: 4x4-1

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 : 4 is the square of 2

Check : 1 is the square of 1

Check : x4 is the square of x2

Factorization is : (2x2 + 1) • (2x2 - 1)

Polynomial Roots Calculator :

2.2 Find roots (zeroes) of : F(x) = 2x2 + 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 2 and the Trailing Constant is 1.

The factor(s) are:

of the Leading Coefficient : 1,2

of the Trailing Constant : 1

Let us test ....

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

-1 1 -1.00 3.00

-1 2 -0.50 1.50

1 1 1.00 3.00

1 2 0.50 1.50

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares:

2.3 Factoring: 2x2 - 1

Check : 2 is not a square !!

Ruling : Binomial can not be factored as the

difference of two perfect squares

Final result :

(2x2 + 1) • (2x2 - 1)

Answer 2

Answer:

(2x²-1)(2x²+1)

Step-by-step explanation:

4x⁴+1

(2x²)²-(-1)²

{2x²+(-1)}{2x²-(-1)}

(2x²-1)(2x²+1)

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