Multiplying a monomial by a polynomial

The distributive law $a(b+c) = ab + ac$ says that multiplying a sum is the same as multiplying each piece and adding. Once you accept this, multiplying a monomial by any polynomial becomes mechanical.

Concept

To multiply a monomial $M$ by a polynomial $P = T_1 + T_2 + \dots + T_n$:

$M \cdot P = M T_1 + M T_2 + \dots + M T_n.$

Each term of the polynomial is multiplied by $M$ separately.

For each multiplication, two things happen:

1. Multiply the coefficients as ordinary numbers, respecting signs.
2. Add the exponents of the same variables, using $x^m \cdot x^n = x^{m+n}$.

Example: $2x^3 \cdot (4x^2 - 5x + 7)$.

• $2x^3 \cdot 4x^2 = 8x^{3+2} = 8x^5$.
• $2x^3 \cdot (-5x) = -10x^4$.
• $2x^3 \cdot 7 = 14x^3$.

Answer: $8x^5 - 10x^4 + 14x^3$.

Signs. The usual rules apply. A positive times a negative is negative; a negative times a negative is positive. If the monomial in front has a minus sign, every term's sign flips.

Example: $-3a \cdot (a^2 - 2a + 5) = -3a^3 + 6a^2 - 15a$.

Multiple variables. Same logic, just multiply each letter's powers separately.

Example: $4xy \cdot (3x - 2y + 1) = 12x^2 y - 8 x y^2 + 4xy$.

Why this matters. Multiplying a monomial across an entire polynomial is the most common step in algebra. Every "expand and simplify" problem starts with this. Whenever you collect like terms after an expansion, you have already used this rule several times.

Geometrically, this is the rule that lets us split a rectangle into smaller rectangles. A rectangle of width $a$ and length $(b + c + d)$ has area $a(b+c+d)$, which equals the sum of the three sub-rectangles $ab + ac + ad$.

Worked examples

Example 1. Expand $5(2x - 7)$.

• $5 \cdot 2x = 10x$.
• $5 \cdot (-7) = -35$.
• Answer: $10x - 35$.

Example 2. Expand $-2y(3y^2 - 5y + 4)$.

• $-2y \cdot 3y^2 = -6y^3$.
• $-2y \cdot (-5y) = 10y^2$.
• $-2y \cdot 4 = -8y$.
• Answer: $-6y^3 + 10y^2 - 8y$.

Example 3. Expand $3a^2 b \cdot (a - 2b + 3ab)$.

• $3a^2 b \cdot a = 3a^3 b$.
• $3a^2 b \cdot (-2b) = -6a^2 b^2$.
• $3a^2 b \cdot 3ab = 9a^3 b^2$.
• Answer: $3a^3 b - 6 a^2 b^2 + 9 a^3 b^2$.

Example 4. Find the area of a rectangle with length $(3x + 4)$ and width $5x$.

• Area $= 5x(3x + 4) = 15x^2 + 20x$.

Try it yourself

1. Expand $7(2a + 3)$.
2. Expand $-4(5x - 6y + 1)$.
3. Expand $2x \cdot (x^2 - 3x + 7)$.
4. Expand $-3a^2 \cdot (a^3 - 2a^2 + a - 1)$.
5. Expand $5xy \cdot (2x^2 - 3xy + y^2)$.
6. A rectangle has length $(2x + 5)$ and width $3$. Express its area as an expression.
7. Simplify $4(x - 2) + 3(2x + 1)$.
8. Simplify $2a(a + b) - 3b(a - b)$.

Activity

Area in pieces. Draw a rectangle on graph paper, labelling its width $a$ and length $b + c + d$. Cut it into three sub-rectangles of widths $b, c, d$ respectively. Count or compute their separate areas, then add. You should get exactly $a(b+c+d)$ , the distributive law in pictures.