Solve 10(b^{2}-2b)^3-20(b^2-2b)^2-30b^2+60b

$$10 ( b ^ { 2 } - 2 b ) ^ { 3 } - 20 ( b ^ { 2 } - 2 b ) ^ { 2 } - 30 b ^ { 2 } + 60 b$$

Evaluate

$$10b\left(b-3\right)\left(b-2\right)\left(b+1\right)\left(b-1\right)^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(b^{2}-2b\right)^{3}.

$$10\left(\left(b^{2}\right)^{3}-6\left(b^{2}\right)^{2}b+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6\left(b^{2}\right)^{2}b+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6b^{4}b+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6b^{5}+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6b^{5}+12b^{4}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

Use the distributive property to multiply 10 by b^{6}-6b^{5}+12b^{4}-8b^{3}.

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(b^{2}-2b\right)^{2}.

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(\left(b^{2}\right)^{2}-4b^{2}b+4b^{2}\right)-30b^{2}+60b$$

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

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(b^{4}-4b^{2}b+4b^{2}\right)-30b^{2}+60b$$

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

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(b^{4}-4b^{3}+4b^{2}\right)-30b^{2}+60b$$

Use the distributive property to multiply -20 by b^{4}-4b^{3}+4b^{2}.

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20b^{4}+80b^{3}-80b^{2}-30b^{2}+60b$$

Combine 120b^{4} and -20b^{4} to get 100b^{4}.

$$10b^{6}-60b^{5}+100b^{4}-80b^{3}+80b^{3}-80b^{2}-30b^{2}+60b$$

Combine -80b^{3} and 80b^{3} to get 0.

$$10b^{6}-60b^{5}+100b^{4}-80b^{2}-30b^{2}+60b$$

Combine -80b^{2} and -30b^{2} to get -110b^{2}.

$$10b^{6}-60b^{5}+100b^{4}-110b^{2}+60b$$

Expand

$$10b^{6}-60b^{5}+100b^{4}-110b^{2}+60b$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(b^{2}-2b\right)^{3}.

$$10\left(\left(b^{2}\right)^{3}-6\left(b^{2}\right)^{2}b+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6\left(b^{2}\right)^{2}b+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6b^{4}b+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6b^{5}+12b^{2}b^{2}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

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

$$10\left(b^{6}-6b^{5}+12b^{4}-8b^{3}\right)-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

Use the distributive property to multiply 10 by b^{6}-6b^{5}+12b^{4}-8b^{3}.

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(b^{2}-2b\right)^{2}-30b^{2}+60b$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(b^{2}-2b\right)^{2}.

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(\left(b^{2}\right)^{2}-4b^{2}b+4b^{2}\right)-30b^{2}+60b$$

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

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(b^{4}-4b^{2}b+4b^{2}\right)-30b^{2}+60b$$

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

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20\left(b^{4}-4b^{3}+4b^{2}\right)-30b^{2}+60b$$

Use the distributive property to multiply -20 by b^{4}-4b^{3}+4b^{2}.

$$10b^{6}-60b^{5}+120b^{4}-80b^{3}-20b^{4}+80b^{3}-80b^{2}-30b^{2}+60b$$

Combine 120b^{4} and -20b^{4} to get 100b^{4}.

$$10b^{6}-60b^{5}+100b^{4}-80b^{3}+80b^{3}-80b^{2}-30b^{2}+60b$$

Combine -80b^{3} and 80b^{3} to get 0.

$$10b^{6}-60b^{5}+100b^{4}-80b^{2}-30b^{2}+60b$$

Combine -80b^{2} and -30b^{2} to get -110b^{2}.

$$10b^{6}-60b^{5}+100b^{4}-110b^{2}+60b$$