Exercises: Prove and Use Polynomial Identities

Work through each section in order. For proof problems, show all algebraic steps clearly.

Grade 9·22 problems·~38 min·Common Core Math - HS Algebra·standard·hsa-apr-c-4

Work through problems with immediate feedback

Fluency Practice

$Two-column proof scaffold showing correct left-to-right transformation of (a−b)² into a²−2ab+b², with an arrow showing it matches the right side.$

To prove the identity $(a - b)^2 = a^2 - 2ab + b^2$ , which sequence of steps is correct?

Start: $(a-b)^2 = a^2 - 2ab + b^2$ . Square-root both sides. Done.

Start from the left side: $(a-b)^2 = (a-b)(a-b) = a^2 - ab - ab + b^2 = a^2 - 2ab + b^2$ . This equals the right side. ✓

Start from both sides simultaneously, add $2ab$ to both sides, and conclude $a^2 = (a-b)^2 + 2ab - b^2$ .

Substitute $a = 1, b = 0$ and verify: $(1-0)^2 = 1$ and $1 - 0 + 0 = 1$ . Identity proved.