Solve 1/2n(n-1)(3n-1)

$$\frac { 1 } { 2 } n ( n - 1 ) ( 3 n - 1 )$$

Evaluate

$$\frac{n\left(n-1\right)\left(3n-1\right)}{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2}n by n-1.

$$\left(\frac{1}{2}nn+\frac{1}{2}n\left(-1\right)\right)\left(3n-1\right)$$

Multiply n and n to get n^{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Apply the distributive property by multiplying each term of \frac{1}{2}n^{2}-\frac{1}{2}n by each term of 3n-1.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Multiply n and n to get n^{2}.

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{3}{2}n^{3}-\frac{1}{2}n^{2}-\frac{1}{2}n^{2}\times 3-\frac{1}{2}n\left(-1\right)$$

Express -\frac{1}{2}\times 3 as a single fraction.

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

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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

Combine -\frac{1}{2}n^{2} and -\frac{3}{2}n^{2} to get -2n^{2}.

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

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$\frac{3}{2}n^{3}-2n^{2}+\frac{1}{2}n$$

Expand

$$\frac{3n^{3}}{2}-2n^{2}+\frac{n}{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{2}n by n-1.

$$\left(\frac{1}{2}nn+\frac{1}{2}n\left(-1\right)\right)\left(3n-1\right)$$

Multiply n and n to get n^{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Apply the distributive property by multiplying each term of \frac{1}{2}n^{2}-\frac{1}{2}n by each term of 3n-1.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

Multiply n and n to get n^{2}.

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{3}{2}n^{3}-\frac{1}{2}n^{2}-\frac{1}{2}n^{2}\times 3-\frac{1}{2}n\left(-1\right)$$

Express -\frac{1}{2}\times 3 as a single fraction.

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

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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

Combine -\frac{1}{2}n^{2} and -\frac{3}{2}n^{2} to get -2n^{2}.

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

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$\frac{3}{2}n^{3}-2n^{2}+\frac{1}{2}n$$