use the polynomial identity to rewrite x^3+a^3=(x+a)(x^2-ax+a^2) to rewrite x^3+25 which polynomial does this establish

1 answer

To rewrite \( x^3 + 25 \) using the polynomial identity \( x^3 + a^3 = (x + a)(x^2 - ax + a^2) \), identify \( a^3 \) in the expression \( x^3 + 25 \).

We can express \( 25 \) as \( 25 = 5^3 \). Therefore, we can set \( a = 5 \).

Using the identity, we have: \[ x^3 + 5^3 = (x + 5)(x^2 - 5x + 5^2) \]

Now substituting \( 5^2 = 25 \): \[ x^3 + 25 = (x + 5)(x^2 - 5x + 25) \]

This establishes the polynomial: \[ x^3 + 25 = (x + 5)(x^2 - 5x + 25) \]

So, the polynomial you are looking for is \( (x + 5)(x^2 - 5x + 25) \).