Solve 4x^6+0x^5+2x^4+0x^3-{x}^2+0x+0divx+1

$$4 { x }^{ 6 } +0 { x }^{ 5 } +2 { x }^{ 4 } +0 { x }^{ 3 } - { x }^{ 2 } +0x+0 \div x+1$$

Evaluate

$$\left(x^{2}+1\right)\left(4x^{4}-2x^{2}+1\right)$$

Solution Steps

Anything times zero gives zero.

$$4x^{6}+0+2x^{4}+0x^{3}-x^{2}+0x+\frac{0}{x}+1$$

Anything times zero gives zero.

$$4x^{6}+0+2x^{4}+0-x^{2}+0x+\frac{0}{x}+1$$

Add 0 and 0 to get 0.

$$4x^{6}+2x^{4}-x^{2}+0x+\frac{0}{x}+1$$

Anything times zero gives zero.

$$4x^{6}+2x^{4}-x^{2}+0+\frac{0}{x}+1$$

Zero divided by any non-zero term gives zero.

$$4x^{6}+2x^{4}-x^{2}+0+0+1$$

Add 0 and 0 to get 0.

$$4x^{6}+2x^{4}-x^{2}+1$$

Factor

$$\left(x^{2}+1\right)\left(4x^{4}-2x^{2}+1\right)$$

Solution Steps

Anything times zero gives zero.

$$factor(4x^{6}+0+2x^{4}+0x^{3}-x^{2}+0x+\frac{0}{x}+1)$$

Anything times zero gives zero.

$$factor(4x^{6}+0+2x^{4}+0-x^{2}+0x+\frac{0}{x}+1)$$

Add 0 and 0 to get 0.

$$factor(4x^{6}+2x^{4}-x^{2}+0x+\frac{0}{x}+1)$$

Anything times zero gives zero.

$$factor(4x^{6}+2x^{4}-x^{2}+0+\frac{0}{x}+1)$$

Zero divided by any non-zero term gives zero.

$$factor(4x^{6}+2x^{4}-x^{2}+0+0+1)$$

Add 0 and 0 to get 0.

$$factor(4x^{6}+2x^{4}-x^{2}+1)$$

Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 4x^{6} and n divides the constant factor 1. One such factor is x^{2}+1. Factor the polynomial by dividing it by this factor. The following polynomials are not factored since they do not have any rational roots: 4x^{4}-2x^{2}+1,x^{2}+1.

$$\left(x^{2}+1\right)\left(4x^{4}-2x^{2}+1\right)$$