Solve f(x)=-1/10(x+3)(x-3)(x+1)^3

$$f ( x ) = - \frac { 1 } { 10 } ( x + 3 ) ( x - 3 ) ( x + 1 ) ^ { 3 }$$

Evaluate

$$-\frac{\left(x^{2}-9\right)\left(x+1\right)^{3}}{10}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$-\frac{1}{10}\left(x+3\right)\left(x-3\right)\left(x^{3}+3x^{2}+3x+1\right)$$

Use the distributive property to multiply -\frac{1}{10} by x+3.

$$\left(-\frac{1}{10}x-\frac{3}{10}\right)\left(x-3\right)\left(x^{3}+3x^{2}+3x+1\right)$$

Use the distributive property to multiply -\frac{1}{10}x-\frac{3}{10} by x-3 and combine like terms.

$$\left(-\frac{1}{10}x^{2}+\frac{9}{10}\right)\left(x^{3}+3x^{2}+3x+1\right)$$

Use the distributive property to multiply -\frac{1}{10}x^{2}+\frac{9}{10} by x^{3}+3x^{2}+3x+1 and combine like terms.

$$-\frac{1}{10}x^{5}-\frac{3}{10}x^{4}+\frac{3}{5}x^{3}+\frac{13}{5}x^{2}+\frac{27}{10}x+\frac{9}{10}$$

Expand

$$-\frac{x^{5}}{10}-\frac{3x^{4}}{10}+\frac{3x^{3}}{5}+\frac{13x^{2}}{5}+\frac{27x}{10}+\frac{9}{10}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(x+1\right)^{3}.

$$-\frac{1}{10}\left(x+3\right)\left(x-3\right)\left(x^{3}+3x^{2}+3x+1\right)$$

Use the distributive property to multiply -\frac{1}{10} by x+3.

$$\left(-\frac{1}{10}x-\frac{3}{10}\right)\left(x-3\right)\left(x^{3}+3x^{2}+3x+1\right)$$

Use the distributive property to multiply -\frac{1}{10}x-\frac{3}{10} by x-3 and combine like terms.

$$\left(-\frac{1}{10}x^{2}+\frac{9}{10}\right)\left(x^{3}+3x^{2}+3x+1\right)$$

Use the distributive property to multiply -\frac{1}{10}x^{2}+\frac{9}{10} by x^{3}+3x^{2}+3x+1 and combine like terms.

$$-\frac{1}{10}x^{5}-\frac{3}{10}x^{4}+\frac{3}{5}x^{3}+\frac{13}{5}x^{2}+\frac{27}{10}x+\frac{9}{10}$$