Solve (1-lambda)(-lambda)(-lambda)(3-lambda)(1-lambda)

$$( 1 - \lambda ) ( - \lambda ) ( - \lambda ) ( 3 - \lambda ) ( 1 - \lambda )$$

Evaluate

$$\left(3-\lambda \right)\left(\lambda \left(1-\lambda \right)\right)^{2}$$

Solution Steps

Multiply 1-\lambda and 1-\lambda to get \left(1-\lambda \right)^{2}.

$$\left(1-\lambda \right)^{2}\left(-\lambda \right)\left(-\lambda \right)\left(3-\lambda \right)$$

Multiply -\lambda and -\lambda to get \left(-\lambda \right)^{2}.

$$\left(1-\lambda \right)^{2}\left(-\lambda \right)^{2}\left(3-\lambda \right)$$

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

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

Calculate -\lambda to the power of 2 and get \lambda ^{2}.

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

Use the distributive property to multiply 1-2\lambda +\lambda ^{2} by \lambda ^{2}.

$$\left(\lambda ^{2}-2\lambda ^{3}+\lambda ^{4}\right)\left(3-\lambda \right)$$

Apply the distributive property by multiplying each term of \lambda ^{2}-2\lambda ^{3}+\lambda ^{4} by each term of 3-\lambda .

$$3\lambda ^{2}-\lambda ^{3}-6\lambda ^{3}+2\lambda ^{4}+3\lambda ^{4}-\lambda ^{5}$$

Combine -\lambda ^{3} and -6\lambda ^{3} to get -7\lambda ^{3}.

$$3\lambda ^{2}-7\lambda ^{3}+2\lambda ^{4}+3\lambda ^{4}-\lambda ^{5}$$

Combine 2\lambda ^{4} and 3\lambda ^{4} to get 5\lambda ^{4}.

$$3\lambda ^{2}-7\lambda ^{3}+5\lambda ^{4}-\lambda ^{5}$$

Expand

$$3\lambda ^{2}-7\lambda ^{3}+5\lambda ^{4}-\lambda ^{5}$$

Solution Steps

Multiply 1-\lambda and 1-\lambda to get \left(1-\lambda \right)^{2}.

$$\left(1-\lambda \right)^{2}\left(-\lambda \right)\left(-\lambda \right)\left(3-\lambda \right)$$

Multiply -\lambda and -\lambda to get \left(-\lambda \right)^{2}.

$$\left(1-\lambda \right)^{2}\left(-\lambda \right)^{2}\left(3-\lambda \right)$$

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

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

Calculate -\lambda to the power of 2 and get \lambda ^{2}.

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

Use the distributive property to multiply 1-2\lambda +\lambda ^{2} by \lambda ^{2}.

$$\left(\lambda ^{2}-2\lambda ^{3}+\lambda ^{4}\right)\left(3-\lambda \right)$$

Apply the distributive property by multiplying each term of \lambda ^{2}-2\lambda ^{3}+\lambda ^{4} by each term of 3-\lambda .

$$3\lambda ^{2}-\lambda ^{3}-6\lambda ^{3}+2\lambda ^{4}+3\lambda ^{4}-\lambda ^{5}$$

Combine -\lambda ^{3} and -6\lambda ^{3} to get -7\lambda ^{3}.

$$3\lambda ^{2}-7\lambda ^{3}+2\lambda ^{4}+3\lambda ^{4}-\lambda ^{5}$$

Combine 2\lambda ^{4} and 3\lambda ^{4} to get 5\lambda ^{4}.

$$3\lambda ^{2}-7\lambda ^{3}+5\lambda ^{4}-\lambda ^{5}$$