To multiply a monomial by a polynomial, use the distributive property together with the product rule for exponents. It is the building block for all polynomial multiplication.
Why it works
The distributive property says $a(b + c) = ab + ac$: a rectangle of width $a$ and length $b + c$ splits into two rectangles, $ab$ and $ac$. With variables, each product multiplies coefficients and adds exponents of the same base, because $x^2 \cdot x^3$ is two factors of $x$ times three more, $x^5$.
Key idea
Multiply the monomial by every term inside the parentheses: multiply the coefficients, add the exponents, and keep each sign.
Worked examples
Worked example: Distribute a monomial
Multiply $-3x^2(2x^2 - 5x + 1)$.
- $-3x^2 \cdot 2x^2 = -6x^4$.
- $-3x^2 \cdot (-5x) = 15x^3$.
- $-3x^2 \cdot 1 = -3x^2$.
Answer: $-6x^4 + 15x^3 - 3x^2$
Worked example: Two variables
Multiply $4ab(3a - 2b)$.
- $4ab \cdot 3a = 12a^2b$ and $4ab \cdot (-2b) = -8ab^2$.
Answer: $12a^2b - 8ab^2$
Worked example: Two products, then combine
Simplify $2x(x - 4) - 3(x^2 + 1)$.
- First product: $2x(x - 4) = 2x^2 - 8x$.
- Second product: the factor is $-3$, sign included: $-3(x^2 + 1) = -3x^2 - 3$.
- Combine like terms: $2x^2 - 3x^2 - 8x - 3 = -x^2 - 8x - 3$.
- Check with $x = 1$: $2(-3) - 3(2) = -12$, and $-1 - 8 - 3 = -12$.
Answer: $-x^2 - 8x - 3$
Keep this picture in mind: multiplying turns side lengths into an area. In the next unit you will run it backward, starting from the area $6x^2 + 9x$ and finding the sides $3x$ and $2x + 3$. That is factoring.
Common mistake
Multiplying the monomial by the first term only, as in $2x(x^2 + 3x + 4) = 2x^3 + 3x + 4$. The area model shows three pieces, so there must be three products. Check with $x = 1$: $2(1 + 3 + 4) = 16$, while $2 + 3 + 4 = 9$.