Solve =4x^4+11x^3-19x^2-44x+12

$$= 4 x ^ { 4 } + 11 x ^ { 3 } - 19 x ^ { 2 } - 44 x + 12$$

Factor

$$\left(x-2\right)\left(4x-1\right)\left(x+2\right)\left(x+3\right)$$

Solution Steps

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

$$4x^{4}+11x^{3}-19x^{2}-44x+12=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 12 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±3,±6,±12,±\frac{3}{2},±1,±2,±4,±\frac{3}{4},±\frac{1}{2},±\frac{1}{4}$$

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.

$$x=2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 4x^{4}+11x^{3}-19x^{2}-44x+12 by x-2 to get 4x^{3}+19x^{2}+19x-6. To factor the result, solve the equation where it equals to 0.

$$4x^{3}+19x^{2}+19x-6=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -6 and q divides the leading coefficient 4. List all candidates \frac{p}{q}.

$$±\frac{3}{2},±3,±6,±\frac{3}{4},±\frac{1}{2},±1,±2,±\frac{1}{4}$$

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.

$$x=-2$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 4x^{3}+19x^{2}+19x-6 by x+2 to get 4x^{2}+11x-3. To factor the result, solve the equation where it equals to 0.

$$4x^{2}+11x-3=0$$

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 4 for a, 11 for b, and -3 for c in the quadratic formula.

$$x=\frac{-11±\sqrt{11^{2}-4\times 4\left(-3\right)}}{2\times 4}$$

Do the calculations.

$$x=\frac{-11±13}{8}$$

Solve the equation 4x^{2}+11x-3=0 when ± is plus and when ± is minus.

$$x=-3 x=\frac{1}{4}$$

Rewrite the factored expression using the obtained roots.

$$\left(x-2\right)\left(4x-1\right)\left(x+2\right)\left(x+3\right)$$

Evaluate

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