Solve N(x)=(6x-7)^3(8x^2+9)^2

$$N ( x ) = ( 6 x - 7 ) ^ { 3 } ( 8 x ^ { 2 } + 9 ) ^ { 2 }$$

Evaluate

$$\left(6x-7\right)^{3}\left(8x^{2}+9\right)^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(6x-7\right)^{3}.

$$\left(216x^{3}-756x^{2}+882x-343\right)\left(8x^{2}+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8x^{2}+9\right)^{2}.

$$\left(216x^{3}-756x^{2}+882x-343\right)\left(64\left(x^{2}\right)^{2}+144x^{2}+81\right)$$

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

$$\left(216x^{3}-756x^{2}+882x-343\right)\left(64x^{4}+144x^{2}+81\right)$$

Use the distributive property to multiply 216x^{3}-756x^{2}+882x-343 by 64x^{4}+144x^{2}+81 and combine like terms.

$$13824x^{7}+87552x^{5}+144504x^{3}-48384x^{6}-130816x^{4}-110628x^{2}+71442x-27783$$

Expand

$$13824x^{7}-48384x^{6}+87552x^{5}-130816x^{4}+144504x^{3}-110628x^{2}+71442x-27783$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(6x-7\right)^{3}.

$$\left(216x^{3}-756x^{2}+882x-343\right)\left(8x^{2}+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(8x^{2}+9\right)^{2}.

$$\left(216x^{3}-756x^{2}+882x-343\right)\left(64\left(x^{2}\right)^{2}+144x^{2}+81\right)$$

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

$$\left(216x^{3}-756x^{2}+882x-343\right)\left(64x^{4}+144x^{2}+81\right)$$

Use the distributive property to multiply 216x^{3}-756x^{2}+882x-343 by 64x^{4}+144x^{2}+81 and combine like terms.

$$13824x^{7}+87552x^{5}+144504x^{3}-48384x^{6}-130816x^{4}-110628x^{2}+71442x-27783$$