Solve -0.00002(x-12)^2(x-17)(x-9)(x-2)(x+4)(x^2+3)

$$- 0.00002 ( x - 12 ) ^ { 2 } ( x - 17 ) ( x - 9 ) ( x - 2 ) ( x + 4 ) ( x ^ { 2 } + 3 )$$

Evaluate

$$-\frac{\left(x-17\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x-12\right)^{2}\left(x^{2}+3\right)}{50000}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$-0.00002\left(x^{2}-24x+144\right)\left(x-17\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002 by x^{2}-24x+144.

$$\left(-0.00002x^{2}+0.00048x-0.00288\right)\left(x-17\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{2}+0.00048x-0.00288 by x-17 and combine like terms.

$$\left(-0.00002x^{3}+0.00082x^{2}-0.01104x+0.04896\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{3}+0.00082x^{2}-0.01104x+0.04896 by x-9 and combine like terms.

$$\left(-0.00002x^{4}+0.001x^{3}-0.01842x^{2}+0.14832x-0.44064\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{4}+0.001x^{3}-0.01842x^{2}+0.14832x-0.44064 by x-2 and combine like terms.

$$\left(-0.00002x^{5}+0.00104x^{4}-0.02042x^{3}+0.18516x^{2}-0.73728x+0.88128\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{5}+0.00104x^{4}-0.02042x^{3}+0.18516x^{2}-0.73728x+0.88128 by x+4 and combine like terms.

$$\left(-0.00002x^{6}+0.00096x^{5}-0.01626x^{4}+0.10348x^{3}+0.00336x^{2}-2.06784x+3.52512\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{6}+0.00096x^{5}-0.01626x^{4}+0.10348x^{3}+0.00336x^{2}-2.06784x+3.52512 by x^{2}+3 and combine like terms.

$$-0.00002x^{8}-0.01632x^{6}+0.00096x^{7}+0.10636x^{5}-0.04542x^{4}-1.7574x^{3}+3.5352x^{2}-6.20352x+10.57536$$

Expand

$$-\frac{x^{8}}{50000}+\frac{3x^{7}}{3125}-\frac{51x^{6}}{3125}+\frac{2659x^{5}}{25000}-\frac{2271x^{4}}{50000}-\frac{8787x^{3}}{5000}+\frac{4419x^{2}}{1250}-\frac{19386x}{3125}+10.57536$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-12\right)^{2}.

$$-0.00002\left(x^{2}-24x+144\right)\left(x-17\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002 by x^{2}-24x+144.

$$\left(-0.00002x^{2}+0.00048x-0.00288\right)\left(x-17\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{2}+0.00048x-0.00288 by x-17 and combine like terms.

$$\left(-0.00002x^{3}+0.00082x^{2}-0.01104x+0.04896\right)\left(x-9\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{3}+0.00082x^{2}-0.01104x+0.04896 by x-9 and combine like terms.

$$\left(-0.00002x^{4}+0.001x^{3}-0.01842x^{2}+0.14832x-0.44064\right)\left(x-2\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{4}+0.001x^{3}-0.01842x^{2}+0.14832x-0.44064 by x-2 and combine like terms.

$$\left(-0.00002x^{5}+0.00104x^{4}-0.02042x^{3}+0.18516x^{2}-0.73728x+0.88128\right)\left(x+4\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{5}+0.00104x^{4}-0.02042x^{3}+0.18516x^{2}-0.73728x+0.88128 by x+4 and combine like terms.

$$\left(-0.00002x^{6}+0.00096x^{5}-0.01626x^{4}+0.10348x^{3}+0.00336x^{2}-2.06784x+3.52512\right)\left(x^{2}+3\right)$$

Use the distributive property to multiply -0.00002x^{6}+0.00096x^{5}-0.01626x^{4}+0.10348x^{3}+0.00336x^{2}-2.06784x+3.52512 by x^{2}+3 and combine like terms.

$$-0.00002x^{8}-0.01632x^{6}+0.00096x^{7}+0.10636x^{5}-0.04542x^{4}-1.7574x^{3}+3.5352x^{2}-6.20352x+10.57536$$