Solve f(x)=5x^4+80x^3+276x^2+16x+55

$$f ( x ) = 5 x ^ { 4 } + 80 x ^ { 3 } + 276 x ^ { 2 } + 16 x + 55$$

Factor

$$\left(x+5\right)\left(x+11\right)\left(5x^{2}+1\right)$$

Solution Steps

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

$$5x^{4}+80x^{3}+276x^{2}+16x+55=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 55 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±11,±55,±\frac{11}{5},±1,±5,±\frac{1}{5}$$

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.

$$x=-5$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 5x^{4}+80x^{3}+276x^{2}+16x+55 by x+5 to get 5x^{3}+55x^{2}+x+11. To factor the result, solve the equation where it equals to 0.

$$5x^{3}+55x^{2}+x+11=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 11 and q divides the leading coefficient 5. List all candidates \frac{p}{q}.

$$±\frac{11}{5},±11,±\frac{1}{5},±1$$

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.

$$x=-11$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide 5x^{3}+55x^{2}+x+11 by x+11 to get 5x^{2}+1. To factor the result, solve the equation where it equals to 0.

$$5x^{2}+1=0$$

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 5 for a, 0 for b, and 1 for c in the quadratic formula.

$$x=\frac{0±\sqrt{0^{2}-4\times 5\times 1}}{2\times 5}$$

Do the calculations.

$$x=\frac{0±\sqrt{-20}}{10}$$

Polynomial 5x^{2}+1 is not factored since it does not have any rational roots.

$$5x^{2}+1$$

Rewrite the factored expression using the obtained roots.

$$\left(x+5\right)\left(x+11\right)\left(5x^{2}+1\right)$$

Evaluate

$$\left(x+5\right)\left(x+11\right)\left(5x^{2}+1\right)$$