Solve (7x + 3b) ^ 3 - (5x + 3b) ^ 3 - 6x(7x + 3b)(5x + 3b) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

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 $0$.

$$8x^{3}$$