Solve -1/2x^3left(x+2right)^2left(x-3right)^2

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

Evaluate

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

Solution Steps

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

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

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

$$-\frac{1}{2}x^{3}\left(x^{2}+4x+4\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply -\frac{1}{2}x^{3} by x^{2}+4x+4.

$$\left(-\frac{1}{2}x^{5}-2x^{4}-2x^{3}\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply -\frac{1}{2}x^{5}-2x^{4}-2x^{3} by x^{2}-6x+9 and combine like terms.

$$-\frac{1}{2}x^{7}+x^{6}+\frac{11}{2}x^{5}-6x^{4}-18x^{3}$$

Expand

$$-\frac{x^{7}}{2}+x^{6}+\frac{11x^{5}}{2}-6x^{4}-18x^{3}$$

Solution Steps

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

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

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

$$-\frac{1}{2}x^{3}\left(x^{2}+4x+4\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply -\frac{1}{2}x^{3} by x^{2}+4x+4.

$$\left(-\frac{1}{2}x^{5}-2x^{4}-2x^{3}\right)\left(x^{2}-6x+9\right)$$

Use the distributive property to multiply -\frac{1}{2}x^{5}-2x^{4}-2x^{3} by x^{2}-6x+9 and combine like terms.

$$-\frac{1}{2}x^{7}+x^{6}+\frac{11}{2}x^{5}-6x^{4}-18x^{3}$$