# Proving Polynomial Identities Worksheets

An identity is an equation that is true for every value of the variable that is present. We will often need to prove or justify that an equation is in fact an identity. There are two methods that we explore here for doing this. I find that students have the greatest rate of success when focusing on just one side of the equation. I encourage them to look at both sides and determine which of the two expression they feel will be easiest to manipulate. Spot that expression and then begin to tinker away at it. A series of worksheets and lessons that help students learn the necessary step to proving that polynomial identities are true.

### Aligned Standard: HSA-APR.C.4

- Finding the Square Step-by-step Lesson- Work on finding the square of the binomial.
- Guided Lesson - Find the square of the polynomial sums and differences.
- Guided Lesson Explanation - I show you how to form polynomials using two simple calculations.
- Practice Worksheet - Evaluate all the binomial squares and simplify them.
- Matching Worksheet - Match the binomials to the polynomials that they create.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Finding the square is what it is all about.

- Homework 1 - Find the square and write your answer in the simplest form.
- Homework 2 - Start with what you know about the square of the binomial.
- Homework 3 - This can usually be done in three steps.

### Practice Worksheets

I find that students have trouble with the square of fractions within operations.

- Practice 1 - When you look under the hood, polynomial identities are really just polynomials that are true.
- Practice 2 - We approach this one from all different levels.
- Practice 3 - Find the end value of these polynomials.

### Math Skill Quizzes

The difficulty level here is very mixed.

- Quiz 1 - We involve simple square roots. think back as to what you can use to counter them.
- Quiz 2 - The value just get larger. Do not get confused when you see multiple variables. Keep them fresh in your mind.
- Quiz 3 - A few more fractions in here. With a cherry (variable) on top.

### How to Prove Polynomial Identities

When you deal with algebra, you deal with various polynomial identities that make simplification a lot easier. Identities are equations that are true for any possible value of the variable. There are several ways we can determine if an equation is an identity. We could examine both sides of the equation and show that they can be changed into the same algebraic expression. We could also focus on one side of the equation and show that it could be transformed into the other side. However, we can prove it also and create new formula as well. In most cases we are checking known or given equations for being identities.

Let us consider an example, where (x^{2} + y^{2})^{2} = (x^{2} - y^{2})^{2} + (2xy) ^{2} is an identity.

Let us take left-hand-side first: (x^{2} + y^{2})^{2}

Let us expand by using the multiplicative and distributive property:

(1) x^{4} + 2x^{2}y^{2} +y^{4} We will leave it here.Let us take Right-hand-side first: (x^{2} - y^{2})^{2} + (2xy)^{2}.

Let us expand by using the multiplicative and distributive property:

(2) x^{4} - 2x2y^{2} +y^{4} + (4x^{2}y^{2})

Now let us put both sides together:

x^{4} + 2x^{2}y^{2} +y^{4} = x^{4} - 2x^{2}y^{2} +y^{4} + (4x^{2}y^{2}). x^{4} + 2x^{2}y^{2} +y^{4} = x^{4} + 2x^{2}y^{2} +y^{4}.

Here, we can conclude that both identities are the same.