Exploring Algebraic Identities

Try to evaluate the following using a suitable identity:

Suppose 7x is split as 2x + 5x; can a similar rectangular arrangement be formed? Consider other possibilities and check.

Try to multiply the following using the distributive property:

Verify this for yourself by choosing different values of x and y. (Context: Thus (x - y)($x^{2}$ + xy + $y^{2}$) = $x^{3}$ - $y^{3}$. This is also an identity!)

Predict what (x + y)($x^{2}$ - xy + $y^{2}$) will be.

Find the value of

Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.

Fill in the blanks with the appropriate expressions to make the equation true. (px + a)(qx + b) = ()$x^{2}$ + ()x + ____. Also, verify your answer using the distributive property.

Use suitable identities to find the following products:

A rectangular pool has area $2x^{2}$ + 7x + 3 square hastas. If its width is 2x + 1 hastas, find its length. Hasta was a unit used to measure length.

Find the values using suitable identities:

Factor the following:

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

Try and find other patterns like this one. For example, you could consider 4 consecutive squares and see if you can find a pattern.

What can you say about a and b if (a + b)^2 < $a^{2}$ + $b^{2}$?

What can you say about a and b if (a + b)^2 > $a^{2}$ + $b^{2}$?

Using the identity (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$, expand the following:

What if we replace b by -b in (a + b)^2 = $a^{2}$ + 2ab + $b^{2}$?

Factor completely:

When will (a + b)^2 be equal to $a^{2}$ + $b^{2}$?