Question

Prove the polynomial identity.
(𝑥+5^)2 =𝑥^2 +10⁢𝑥 +25

(𝑥+5)^2 ? = 𝑥^2 +10⁢𝑥 +25 Original equation

𝑥^2 +10⁢𝑥 +25 =𝑥^2 +10⁢𝑥 +25 - distributive property, difference of squares, or square all terms

Answer 1

We want to show (x+5)^2 = x^2 + 10x + 25 for all x.

Expand using the distributive property (or FOIL):
(x+5)^2 = (x+5)(x+5)
= x(x+5) + 5(x+5)
= x^2 + 5x + 5x + 25
= x^2 + 10x + 25.

Equivalently, use the binomial square formula (a+b)^2 = a^2 + 2ab + b^2 with a = x, b = 5:
(x+5)^2 = x^2 + 2·x·5 + 5^2 = x^2 + 10x + 25.

Subtracting the right-hand side from the left gives the zero polynomial, so the equality holds for every x.