Simplifying Algebraic Expressions

This lesson works through several simplification problems, expanding products of algebraic expressions and collecting like terms to reach the simplest form.

Product and sum of two expressions

Simplify $(x+y)(2x+y) + (x+2y)(x-y)$.

First product

$$(x+y)(2x+y) = 2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2$$

Second product

$$(x+2y)(x-y) = x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2$$

Adding the two and collecting like terms:

$$3x^2 + 4xy - y^2$$

Expanding three terms times three terms

Simplify $(a+b+c)(a+b-c)$. Multiply the second bracket by each term of the first:

$$(a+b+c)(a+b-c) = a^2 + ab - ac + ab + b^2 - bc + ac + bc - c^2$$

The terms $-ac$ and $+ac$ cancel, and $-bc$ and $+bc$ cancel, leaving:

$$a^2 + b^2 - c^2 + 2ab$$

Combining two products

Simplify $(a+b)(c-d) + (a-b)(c+d)$. Note the sign in front of each bracket as you expand.

First product

$$(a+b)(c-d) = ac - ad + bc - bd$$

Second product

$$(a-b)(c+d) = ac + ad - bc - bd$$

Adding them, $-ad$ and $+ad$ cancel, and $+bc$ and $-bc$ cancel:

$$2ac - 2bd$$

Subtracting a product from another

Simplify $(a+b)(2a-3b+c) - (2a-3b)c$.

First product

$$(a+b)(2a-3b+c) = 2a^2 - 3ab + ac + 2ab - 3b^2 + bc$$

Subtracted product

$$-(2a-3b)c = -2ac + 3bc$$

Collecting like terms, the square terms first:

$$2a^2 - 3b^2 - ab - ac + 4bc$$

Here $-3ab + 2ab = -ab$, $ac - 2ac = -ac$, and $bc + 3bc = 4bc$.

Binomial times a trinomial

Simplify $(a+7)(a^2 + 3a + 5)$. Multiply the trinomial by each term of the binomial:

$$(a+7)(a^2+3a+5) = a^3 + 3a^2 + 5a + 7a^2 + 21a + 35$$

Collecting like terms:

$$a^3 + 10a^2 + 26a + 35$$

Key takeaways

• Expand by multiplying every term of one bracket by every term of the other.
• When a minus sign sits in front of a bracket, it changes the sign of every term inside.
• Collect like terms at the end so opposite terms cancel and the expression simplifies.