While an equation might be true for some specific value of variables within it, a mathematical identity is true for every value of the variables within it. Many of the patterns used in factoring polynomials are mathematical identities.

Press the 'Play/Pause' button to progress through the slides. Or use the slider to change the slides more slowly.

Use your observations to answer the questions.

In slide 1, what is the area of the red square in terms of a?

In slide 2, what is the area of the blue square in terms of b?

If the blue square is taken away, what expression can we use to represent the area that remains?

In slides 3 and 4, what's left of the square is rearranged to create a rectangle. Write the dimensions of the new rectangle in terms of a and b.

Write the area of the new rectangle in terms of a and b.

Explain how the equation you found in question 3 is equal to the area of the rectangle you found in question 5.

One mathematical identity we have worked with already is the **difference of squares**, m^{2}-n^{2}=\left(m-n\right)\left(m+n\right). It is easily proven using a geometric diagram such as the one from the exploration.

Another familiar identity we have previously used is {a^{2} + 2 a b + b^{2} = \left(a + b\right)^{2}} when factoring **perfect square trinomials**. Two more important identities are the **sum of cubes ** and **difference of cubes**.

In the proof of an identity, it is our job to prove both sides are equal, so we must work with one side of the equation and show that algebraic manipulation leads to the other side. We cannot manipulate both sides of the equation at once because changing both sides assumes that both sides are already equal.

The perfect square trinomial identity is \left(a+b\right)^{2}=a^{2}+2ab+b^{2}.

a

Prove the identity.

Worked Solution

b

Use the identity to evaluate 98^{2}.

Worked Solution

Use an identity to expand the expression \left(x - 6\right) \left(x^{2} + 6 x + 36\right). State which identity was used.

Worked Solution

Factor the following expressions and identify which identity was used.

a

25r^{2} - 60rs + 36s^{2}

Worked Solution

b

64m^{6} - 81n^{4}

Worked Solution

c

125g^{3}+64h^{3}

Worked Solution

Use the algebra tiles given to verify that x^{2}-6x+9=\left(x-3\right)^{2}.

Worked Solution

Idea summary

Important identities we use often are:

Perfect square trinomials: a^{2} + 2 a b + b^{2} = \left(a + b\right)^{2} \text{ or } a^{2} - 2 a b + b^{2} = \left(a - b\right)^{2}

Difference of squares: a^{2} - b^{2} = \left(a+b\right)\left(a-b\right)

Sum of cubes: a^{3}+b^{3}=\left(a+b\right)\left(a^{2}-ab+b^{2}\right)

Difference of cubes: a^{3}-b^{3}=\left(a-b\right)\left(a^{2}+ab+b^{2}\right)

We can verify identities mathematically through algebraic manipulation or using geometric diagrams. These identities can be used to describe numerical relationships.