Solve f(x)=(3x^2-5x)^3

$$f ( x ) = ( 3 x ^ { 2 } - 5 x ) ^ { 3 }$$

Evaluate

$$\left(x\left(3x-5\right)\right)^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(3x^{2}-5x\right)^{3}.

$$27\left(x^{2}\right)^{3}-135\left(x^{2}\right)^{2}x+225x^{2}x^{2}-125x^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$27x^{6}-135\left(x^{2}\right)^{2}x+225x^{2}x^{2}-125x^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$27x^{6}-135x^{4}x+225x^{2}x^{2}-125x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$27x^{6}-135x^{5}+225x^{2}x^{2}-125x^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$27x^{6}-135x^{5}+225x^{4}-125x^{3}$$

Expand

$$27x^{6}-135x^{5}+225x^{4}-125x^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(3x^{2}-5x\right)^{3}.

$$27\left(x^{2}\right)^{3}-135\left(x^{2}\right)^{2}x+225x^{2}x^{2}-125x^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$27x^{6}-135\left(x^{2}\right)^{2}x+225x^{2}x^{2}-125x^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$27x^{6}-135x^{4}x+225x^{2}x^{2}-125x^{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$27x^{6}-135x^{5}+225x^{2}x^{2}-125x^{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$27x^{6}-135x^{5}+225x^{4}-125x^{3}$$