A monomial is an expression with only one term (e.g., $5x$ or $3y$). When multiplying two or more monomials, the rule is simple: multiply the numerical coefficients first, then multiply the variables.
$5x \times 3y$
Multiply coefficients: $5 \times 3 = 15$
Multiply variables: $x \times y = xy$
Result: $15xy$
$5x \times (-4xyz)$
Multiply coefficients: $5 \times (-4) = -20$
Multiply variables: $x \times x \times y \times z = x^2yz$
Result: $-20x^2yz$
$4xy \times 5x^2y^2 \times 6x^3y^3$
Multiply coefficients: $4 \times 5 \times 6 = 120$
Multiply variables: Sum the exponents of like variables ($x^{1+2+3}$ and $y^{1+2+3}$)
Result: $120x^6y^6$
When you multiply a single term (monomial) by an expression with multiple terms (polynomial), you use the Distributive Property. This means you multiply the outer term by every term inside the parentheses.
$a(b + c) = ab + ac$
$a(b – c) = ab – ac$
$a(b + c + d) = ab + ac + ad$
Find the product of $2x(3x + 5xy)$
$2x \times 3x = 6x^2$
$2x \times 5xy = 10x^2y$
Solution: $6x^2 + 10x^2y$
Find the product of $a^2(2ab – 5c)$
$a^2 \times 2ab = 2a^3b$
$a^2 \times (-5c) = -5a^2c$
Solution: $2a^3b – 5a^2c$
Multiplying two binomials (polynomials with two terms) follows a specific pattern often referred to as the FOIL method (First, Outer, Inner, Last). Essentially, every term in the first set of parentheses must be multiplied by every term in the second set.
$(a + b)(c + d) = ac + ad + bc + bd$
$(a + b)(c – d) = ac – ad + bc – bd$
$(a – b)(c + d) = ac + ad – bc – bd$
$(a – b)(c – d) = ac – ad – bc + bd$
Pay close attention to the signs (+ or -). Multiplying two negatives results in a positive, while multiplying a positive and a negative results in a negative.
Multiplying two polynomials requires multiplying every term in the first expression by every term in the second. This often results in a series of terms that must then be simplified by combining “like terms” (terms with the same variable and exponent).
Example: $(x – 4)(2x + 3)$
Multiply terms: $x(2x) + x(3) – 4(2x) – 4(3) = 2x^2 + 3x – 8x – 12$.
Combine like terms ($3x – 8x$): $2x^2 – 5x – 12$.
Example: $(a^2 + 2b^2)(5a – 3b)$
Multiply terms: $5a^3 – 3a^2b + 10ab^2 – 6b^3$.
When expressions involve multiple multiplications and subtractions, it is vital to distribute terms carefully, especially when a negative sign is involved.
Example: $(a + b)(2a – 3b + c) – (2a – 3b)c$
Expand the first part: $2a^2 – 3ab + ac + 2ab – 3b^2 + bc$.
Expand the second part (distribute $-c$): $-2ac + 3bc$.
Combine all terms and simplify: $2a^2 – ab – ac – 3b^2 + 4bc$.
Sometimes you must first find the product of terms before adding or subtracting them from another expression.
$3p(4p^2 + 5p + 7)$
Distribute $3p$: $12p^3 + 15p^2 + 21p$.
Add $5m(3 – m)$ and $6m^3 – 13m$
First, expand: $15m – 5m^2$.
Then add: $(15m – 5m^2) + (6m^3 – 13m)$.
Final result: $6m^3 – 5m^2 + 2m$.
Subtract $3pq(p – q)$ from $2pq(p + q)$
Set up the expression: $2pq(p + q) – 3pq(p – q)$.
Expand both parts: $(2p^2q + 2pq^2) – (3p^2q – 3pq^2)$.
Distribute the negative and simplify: $2p^2q + 2pq^2 – 3p^2q + 3pq^2 = -p^2q + 5pq^2$.
When multiplying binomials or trinomials, every term in the first expression must be multiplied by every term in the second.
Example: $(x – 4)(2x + 3)$ * First, expand the terms: $2x^2 + 3x – 8x – 12$.
Simplify by combining the like terms ($3x$ and $-8x$): $2x^2 – 5x – 12$.
Example: $(a^2 + 2b^2)(5a – 3b)$
Expanding results in: $5a^3 – 3a^2b + 10ab^2 – 6b^3$.
In many problems, you will need to perform multiplication first and then add or subtract the results from other algebraic terms.
Adding Expressions: To add $5m(3 – m)$ and $6m^3 – 13m$, first distribute the $5m$ to get $15m – 5m^2$. Combining this with the second expression gives: $6m^3 – 5m^2 + 2m$.
Subtracting Expressions: When subtracting $3pq(p – q)$ from $2pq(p + q)$, distribute both sets first.
$(2p^2q + 2pq^2) – (3p^2q – 3pq^2)$.
Be careful with the negative sign; it changes the $-3pq^2$ to $+3pq^2$, resulting in: $-p^2q + 5pq^2$.
When an expression has multiple types of brackets (parentheses (), braces {}, and square brackets []), always work from the innermost set outward.
Example: $a – [2b – \{3a – (2b – 3c)\}]$
Remove parentheses: $a – [2b – \{3a – 2b + 3c\}]$.
Remove braces: $a – [2b – 3a + 2b – 3c]$.
Simplify inside brackets: $a – [4b – 3a – 3c]$.
Final result: $4a – 4b + 3c$.
One of the most common mistakes in algebra is “dropping” a negative sign. As seen in the simplification of $(a^2 + b^2 + 2ab) – (a^2 + b^2 – 2ab)$, distributing the negative sign across the second group changes the signs of every term inside, ultimately simplifying the whole expression down to $4ab$.