Solve (7x+3b)^3-(5x+3b)^3-6x(7x+3b)(5x+3b)

$$( 7 x + 3 b ) ^ { 3 } - ( 5 x + 3 b ) ^ { 3 } - 6 x ( 7 x + 3 b ) ( 5 x + 3 b )$$

Evaluate

$$8x^{3}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(7x+3b\right)^{3}.

$$343x^{3}+441x^{2}b+189xb^{2}+27b^{3}-\left(5x+3b\right)^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(5x+3b\right)^{3}.

$$343x^{3}+441x^{2}b+189xb^{2}+27b^{3}-\left(125x^{3}+225x^{2}b+135xb^{2}+27b^{3}\right)-6x\left(7x+3b\right)\left(5x+3b\right)$$

To find the opposite of 125x^{3}+225x^{2}b+135xb^{2}+27b^{3}, find the opposite of each term.

$$343x^{3}+441x^{2}b+189xb^{2}+27b^{3}-125x^{3}-225x^{2}b-135xb^{2}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 343x^{3} and -125x^{3} to get 218x^{3}.

$$218x^{3}+441x^{2}b+189xb^{2}+27b^{3}-225x^{2}b-135xb^{2}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 441x^{2}b and -225x^{2}b to get 216x^{2}b.

$$218x^{3}+216x^{2}b+189xb^{2}+27b^{3}-135xb^{2}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 189xb^{2} and -135xb^{2} to get 54xb^{2}.

$$218x^{3}+216x^{2}b+54xb^{2}+27b^{3}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 27b^{3} and -27b^{3} to get 0.

$$218x^{3}+216x^{2}b+54xb^{2}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Use the distributive property to multiply -6x by 7x+3b.

$$218x^{3}+216x^{2}b+54xb^{2}+\left(-42x^{2}-18xb\right)\left(5x+3b\right)$$

Use the distributive property to multiply -42x^{2}-18xb by 5x+3b and combine like terms.

$$218x^{3}+216x^{2}b+54xb^{2}-210x^{3}-216x^{2}b-54xb^{2}$$

Combine 218x^{3} and -210x^{3} to get 8x^{3}.

$$8x^{3}+216x^{2}b+54xb^{2}-216x^{2}b-54xb^{2}$$

Combine 216x^{2}b and -216x^{2}b to get 0.

$$8x^{3}+54xb^{2}-54xb^{2}$$

Combine 54xb^{2} and -54xb^{2} to get 0.

$$8x^{3}$$

Expand

$$8x^{3}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(7x+3b\right)^{3}.

$$343x^{3}+441x^{2}b+189xb^{2}+27b^{3}-\left(5x+3b\right)^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(5x+3b\right)^{3}.

$$343x^{3}+441x^{2}b+189xb^{2}+27b^{3}-\left(125x^{3}+225x^{2}b+135xb^{2}+27b^{3}\right)-6x\left(7x+3b\right)\left(5x+3b\right)$$

To find the opposite of 125x^{3}+225x^{2}b+135xb^{2}+27b^{3}, find the opposite of each term.

$$343x^{3}+441x^{2}b+189xb^{2}+27b^{3}-125x^{3}-225x^{2}b-135xb^{2}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 343x^{3} and -125x^{3} to get 218x^{3}.

$$218x^{3}+441x^{2}b+189xb^{2}+27b^{3}-225x^{2}b-135xb^{2}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 441x^{2}b and -225x^{2}b to get 216x^{2}b.

$$218x^{3}+216x^{2}b+189xb^{2}+27b^{3}-135xb^{2}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 189xb^{2} and -135xb^{2} to get 54xb^{2}.

$$218x^{3}+216x^{2}b+54xb^{2}+27b^{3}-27b^{3}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Combine 27b^{3} and -27b^{3} to get 0.

$$218x^{3}+216x^{2}b+54xb^{2}-6x\left(7x+3b\right)\left(5x+3b\right)$$

Use the distributive property to multiply -6x by 7x+3b.

$$218x^{3}+216x^{2}b+54xb^{2}+\left(-42x^{2}-18xb\right)\left(5x+3b\right)$$

Use the distributive property to multiply -42x^{2}-18xb by 5x+3b and combine like terms.

$$218x^{3}+216x^{2}b+54xb^{2}-210x^{3}-216x^{2}b-54xb^{2}$$

Combine 218x^{3} and -210x^{3} to get 8x^{3}.

$$8x^{3}+216x^{2}b+54xb^{2}-216x^{2}b-54xb^{2}$$

Combine 216x^{2}b and -216x^{2}b to get 0.

$$8x^{3}+54xb^{2}-54xb^{2}$$

Combine 54xb^{2} and -54xb^{2} to get 0.

$$8x^{3}$$