Solve (5x^{4}-10x^3+3x^2-4)+(3x^4-6x^2+3)

$$( 5 x ^ { 4 } - 10 x ^ { 3 } + 3 x ^ { 2 } - 4 ) + ( 3 x ^ { 4 } - 6 x ^ { 2 } + 3 )$$

Evaluate

$$8x^{4}-10x^{3}-3x^{2}-1$$

Solution Steps

Combine 5x^{4} and 3x^{4} to get 8x^{4}.

$$8x^{4}-10x^{3}+3x^{2}-4-6x^{2}+3$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$8x^{4}-10x^{3}-3x^{2}-4+3$$

Add -4 and 3 to get -1.

$$8x^{4}-10x^{3}-3x^{2}-1$$

Factor

$$\left(2x+1\right)\left(4x^{3}-7x^{2}+2x-1\right)$$

Solution Steps

Multiply and combine like terms.

$$8x^{4}-10x^{3}-3x^{2}-1$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1 and q divides the leading coefficient 8. One such root is -\frac{1}{2}. Factor the polynomial by dividing it by 2x+1. Polynomial 4x^{3}-7x^{2}+2x-1 is not factored since it does not have any rational roots.

$$\left(2x+1\right)\left(4x^{3}-7x^{2}+2x-1\right)$$