What identity can be used to rewrite the expression 8x^3+243
Accepted Solution
A:
To rewrite this expression we can use the sum of cubes identity: [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]. Notice that we can express 8 as a cube: [tex]8=2*2*2=2^3[/tex], so we can rewrite our first term as [tex](2x)^3[/tex]. Since our second term does not have a exact cubic root, we must rewrite as [tex] \sqrt[3]{243} ^{3}[/tex]. Now we have [tex]a=2x[/tex] and [tex]b= \sqrt[3]{243} [/tex], so lets use the sum of cube identity to rewrite our expression: [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex] [tex](2x)^3+ \sqrt[3]{243} ^3=(2x+ \sqrt[3]{243} )((2x)^2-2x \sqrt[3]{243} + \sqrt[3]{243} ^2[/tex] [tex](2x)^3+ \sqrt[3]{243} ^3=(2x+ 3\sqrt[3]{3^2} )((4x^2-6x \sqrt[3]{3^2} +27 \sqrt[3]{3} )[/tex] [tex](2x)^3+ \sqrt[3]{243} ^3=(2x+ 3\sqrt[3]{9} )((4x^2-6x \sqrt[3]{9} +27 \sqrt[3]{3} ) [/tex]
We can conclude that we can use the sum of cubes identity to rewrite the expression 8x^3+243 as [tex](2x+ 3\sqrt[3]{9} )((4x^2-6x \sqrt[3]{9} +27 \sqrt[3]{3} )[/tex]