Question

What identity can be used to rewrite the expression 8x^3+243

Answer 1

Answer: Sum of Cubes

Explanation:

A

P

E

X

Answer 2

To rewrite this expression we can use the sum of cubes identity:
`a^3+b^3=(a+b)(a^2-ab+b^2)`. Notice that we can express 8 as a cube:
`8=2*2*2=2^3`, so we can rewrite our first term as
`(2x)^3`. Since our second term does not have a exact cubic root, we must rewrite as
`\sqrt[3]{243} ^(3)`. Now we have
`a=2x` and
`b= \sqrt[3]{243}`, so lets use the sum of cube identity to rewrite our expression:

`a^3+b^3=(a+b)(a^2-ab+b^2)`

`(2x)^3+ \sqrt[3]{243} ^3=(2x+ \sqrt[3]{243} )((2x)^2-2x \sqrt[3]{243} + \sqrt[3]{243} ^2`

`(2x)^3+ \sqrt[3]{243} ^3=(2x+ 3\sqrt[3]{3^2} )((4x^2-6x \sqrt[3]{3^2} +27 \sqrt[3]{3} )`

`(2x)^3+ \sqrt[3]{243} ^3=(2x+ 3\sqrt[3]{9} )((4x^2-6x \sqrt[3]{9} +27 \sqrt[3]{3} )`

We can conclude that we can use the sum of cubes identity to rewrite the expression 8x^3+243 as
`(2x+ 3\sqrt[3]{9} )((4x^2-6x \sqrt[3]{9} +27 \sqrt[3]{3} )`