Solve 2x^5-7x^3-4x^2-3x^3

$$2 { x }^{ 5 } -7 { x }^{ 3 } -4 { x }^{ 2 } -3 { x }^{ 3 }$$

Factor

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

Solution Steps

Factor out x^{2}.

$$x^{2}\left(2x^{3}-7x-4-3x\right)$$

Consider 2x^{3}-7x-4-3x. Multiply and combine like terms.

$$2x^{3}-10x-4$$

Consider 2x^{3}-10x-4. Factor out 2.

$$2\left(x^{3}-5x-2\right)$$

Consider x^{3}-5x-2. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -2 and q divides the leading coefficient 1. One such root is -2. Factor the polynomial by dividing it by x+2.

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

Rewrite the complete factored expression. Polynomial x^{2}-2x-1 is not factored since it does not have any rational roots.

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

Evaluate

$$2x^{2}\left(x^{3}-5x-2\right)$$

Solution Steps

Combine -7x^{3} and -3x^{3} to get -10x^{3}.

$$2x^{5}-10x^{3}-4x^{2}$$