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

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

Factor

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

Solution Steps

To factor the expression, solve the equation where it equals to 0.

$$2x^{4}+x^{3}+7x^{2}+4x-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±2,±4,±1,±\frac{1}{2}$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{4}+x^{3}+7x^{2}+4x-4 by x+1 to get 2x^{3}-x^{2}+8x-4. To factor the result, solve the equation where it equals to 0.

$$2x^{3}-x^{2}+8x-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -4 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.

$$±2,±4,±1,±\frac{1}{2}$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=\frac{1}{2}$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 2x^{3}-x^{2}+8x-4 by 2\left(x-\frac{1}{2}\right)=2x-1 to get x^{2}+4. To factor the result, solve the equation where it equals to 0.

$$x^{2}+4=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 0 for b, and 4 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$x=\frac{0±\sqrt{-16}}{2}$$

Polynomial x^{2}+4 is not factored since it does not have any rational roots.

$$x^{2}+4$$

Rewrite the factored expression using the obtained roots.

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

Evaluate

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