Multiplication of Algebraic Terms

This lesson covers how to multiply algebraic terms. The method is always the same: fix the sign, multiply the number coefficients, then add the exponents of each repeated variable.

Sign rules

When multiplying signed numbers, the sign of the product follows four rules:

$(+)\times(+) = +,\quad (-)\times(-) = +,\quad (+)\times(-) = -,\quad (-)\times(+) = -$

Laws of exponents

Multiplying powers of the same base uses these two results:

$a^{m}\times a^{n} = a^{m+n}$

$\left(a^{m}\right)^{n} = a^{mn}$

The first rule, adding exponents, is the one used throughout these examples.

Example 1: $8x^{2}y \times 3xy^{2}$

Multiply the numbers, then group each variable.

$8x^{2}y \times 3xy^{2} = (8\times 3)\,(x^{2}\times x)\,(y\times y^{2})$

Numbers: $8\times 3 = 24$. Variable $x$: $x^{2+1} = x^{3}$. Variable $y$: $y^{1+2} = y^{3}$.

$= 24x^{3}y^{3}$

Example 2: $4pq \times 8p^{2}q^{2} \times 3p$

$4pq \times 8p^{2}q^{2} \times 3p = (4\times 8\times 3)\,p^{1+2+1}\,q^{1+2}$

Numbers: $4\times 8\times 3 = 96$. Variable $p$: $p^{1+2+1} = p^{4}$. Variable $q$: $q^{1+2} = q^{3}$.

$= 96p^{4}q^{3}$

Example 3: $(-3x^{2}y^{2})\times(-2x^{3}y)\times(-4xy^{3})$

First the sign: three negative factors give a negative product, since $(-)\times(-) = +$ and then $(+)\times(-) = -$.

$(-3x^{2}y^{2})(-2x^{3}y)(-4xy^{3}) = -(3\times 2\times 4)\,x^{2+3+1}\,y^{2+1+3}$

Numbers: $3\times 2\times 4 = 24$. Variable $x$: $x^{2+3+1} = x^{6}$. Variable $y$: $y^{2+1+3} = y^{6}$.

$= -24x^{6}y^{6}$

Example 4: $800x^{3}y^{2}z \times (-2xy^{5}z^{2}) \times 1$

$800x^{3}y^{2}z \times (-2xy^{5}z^{2}) \times 1 = (800\times(-2)\times 1)\,x^{3+1}\,y^{2+5}\,z^{1+2}$

Numbers: $800\times(-2)\times 1 = -1600$. Variable $x$: $x^{3+1} = x^{4}$. Variable $y$: $y^{2+5} = y^{7}$. Variable $z$: $z^{1+2} = z^{3}$.

$= -1600x^{4}y^{7}z^{3}$

Key takeaways

• The sign of a product is positive when there is an even number of negative factors and negative when there is an odd number.
• Multiply the numerical coefficients of all the terms together.
• For each variable, add the exponents from every term in which it appears.