Solve (x-1)(x+5)(x+8)(x+2)-880=0

$$( x - 1 ) ( x + 5 ) ( x + 8 ) ( x + 2 ) - 880 = 0$$

Solve for x (complex solution)

$$x=-10$$

$$x=3$$

$$x=\frac{-\sqrt{79}i-7}{2}\approx -3.5-4.444097209i$$

$$x=\frac{-7+\sqrt{79}i}{2}\approx -3.5+4.444097209i$$

Solution Steps

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)-880=0$$

Multiply and combine like terms.

$$x^{4}+14x^{3}+51x^{2}+14x-960=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -960 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±960,±480,±320,±240,±192,±160,±120,±96,±80,±64,±60,±48,±40,±32,±30,±24,±20,±16,±15,±12,±10,±8,±6,±5,±4,±3,±2,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+14x^{3}+51x^{2}+14x-960 by x-3 to get x^{3}+17x^{2}+102x+320. Solve the equation where the result equals to 0.

$$x^{3}+17x^{2}+102x+320=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 320 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±320,±160,±80,±64,±40,±32,±20,±16,±10,±8,±5,±4,±2,±1$$

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.

$$x=-10$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+17x^{2}+102x+320 by x+10 to get x^{2}+7x+32. Solve the equation where the result equals to 0.

$$x^{2}+7x+32=0$$

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, 7 for b, and 32 for c in the quadratic formula.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\times 32}}{2}$$

Do the calculations.

$$x=\frac{-7±\sqrt{-79}}{2}$$

Solve the equation x^{2}+7x+32=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{79}i-7}{2} x=\frac{-7+\sqrt{79}i}{2}$$

List all found solutions.

$$x=3 x=-10 x=\frac{-\sqrt{79}i-7}{2} x=\frac{-7+\sqrt{79}i}{2}$$

Solve for x

$$x=-10$$

$$x=3$$

Solution Steps

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$\left(x-1\right)\left(x+2\right)\left(x+5\right)\left(x+8\right)-880=0$$

Multiply and combine like terms.

$$x^{4}+14x^{3}+51x^{2}+14x-960=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -960 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±960,±480,±320,±240,±192,±160,±120,±96,±80,±64,±60,±48,±40,±32,±30,±24,±20,±16,±15,±12,±10,±8,±6,±5,±4,±3,±2,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}+14x^{3}+51x^{2}+14x-960 by x-3 to get x^{3}+17x^{2}+102x+320. Solve the equation where the result equals to 0.

$$x^{3}+17x^{2}+102x+320=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 320 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±320,±160,±80,±64,±40,±32,±20,±16,±10,±8,±5,±4,±2,±1$$

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.

$$x=-10$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+17x^{2}+102x+320 by x+10 to get x^{2}+7x+32. Solve the equation where the result equals to 0.

$$x^{2}+7x+32=0$$

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, 7 for b, and 32 for c in the quadratic formula.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\times 32}}{2}$$

Do the calculations.

$$x=\frac{-7±\sqrt{-79}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$x\in \emptyset$$

List all found solutions.

$$x=3 x=-10$$