Solve x^4-1/2x^2-19/2x^2+9/2x+9/2

$${ x }^{ 4 } - \frac{ 1 }{ 2 } { x }^{ 2 } - \frac{ 19 }{ 2 } { x }^{ 2 } + \frac{ 9 }{ 2 } x+ \frac{ 9 }{ 2 }$$

Factor

$$\frac{\left(x-1\right)\left(2x^{3}+2x^{2}-18x-9\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{2x^{4}-x^{2}-19x^{2}+9x+9}{2}$$

Consider 2x^{4}-x^{2}-19x^{2}+9x+9. Multiply and combine like terms.

$$2x^{4}-20x^{2}+9x+9$$

Consider 2x^{4}-20x^{2}+9x+9. By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 9 and q divides the leading coefficient 2. One such root is 1. Factor the polynomial by dividing it by x-1.

$$\left(x-1\right)\left(2x^{3}+2x^{2}-18x-9\right)$$

Rewrite the complete factored expression. Polynomial 2x^{3}+2x^{2}-18x-9 is not factored since it does not have any rational roots.

$$\frac{\left(x-1\right)\left(2x^{3}+2x^{2}-18x-9\right)}{2}$$

Evaluate

$$x^{4}-10x^{2}+\frac{9x}{2}+\frac{9}{2}$$

Solution Steps

Combine -\frac{1}{2}x^{2} and -\frac{19}{2}x^{2} to get -10x^{2}.

$$x^{4}-10x^{2}+\frac{9}{2}x+\frac{9}{2}$$