Multiplying Algebraic Expressions (Part 1)

This lesson shows how to multiply algebraic expressions, working from single terms up to multiplying polynomials. The rule throughout is the same: deal with the number coefficients and signs first, then combine the variables by adding the powers of matching letters, and finally collect any like terms.

Multiplying monomials

When multiplying monomials, multiply the numbers (with their signs) and multiply the variable parts by adding the exponents of equal bases.

$5a \cdot 3a \cdot 7a^2 = (5 \cdot 3 \cdot 7)\, a^{1+1+2} = 105a^4$

$4y \cdot 8y^2 \cdot y = (4 \cdot 8 \cdot 1)\, y^{1+2+1} = 32y^4$

A term times a bracket

Multiply the outside term into every term inside the bracket.

$3y(2y - 7) = 3y \cdot 2y - 3y \cdot 7 = 6y^2 - 21y$

$2(x^2 - 4y^2 + 5) = 2x^2 - 8y^2 + 10$

$x(x - 3) + 2 = x^2 - 3x + 2$

Fractional and negative coefficients

Multiply the numbers first, cancelling common factors, then the variables. Here $\tfrac{2}{3}$ and $-\tfrac{9}{10}$ give $-\tfrac{3}{5}$ after cancelling.

$\tfrac{2}{3}xy \cdot \left(-\tfrac{9}{10}x^2y^2\right) = -\tfrac{3}{5}\, x^{3} y^{3}$

Multiplying two binomials

Each term of the first binomial multiplies each term of the second, then collect like terms.

$(x + 4)(2x + 3) = 2x^2 + 3x + 8x + 12 = 2x^2 + 11x + 12$

$(x - y)(3x + 5y) = 3x^2 + 5xy - 3xy - 5y^2 = 3x^2 + 2xy - 5y^2$

A binomial times a trinomial

Multiply each term of the binomial across all three terms of the trinomial, then gather like terms.

$(a + 7)(a^2 + 3a + 5)$

First term: $a(a^2 + 3a + 5) = a^3 + 3a^2 + 5a$

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

$a^3 + 3a^2 + 7a^2 + 5a + 21a + 35 = a^3 + 10a^2 + 26a + 35$

Mixed coefficients example

First distribute the $4$ into the second bracket, then multiply the two brackets and collect like terms.

$\left(\tfrac{3}{4}a^2 + 3b^2\right) \cdot 4\left(a^2 - \tfrac{2}{3}b^2\right)$

$\left(\tfrac{3}{4}a^2 + 3b^2\right)\left(4a^2 - \tfrac{8}{3}b^2\right) = 3a^4 - 2a^2b^2 + 12a^2b^2 - 8b^4 = 3a^4 + 10a^2b^2 - 8b^4$

Key takeaways

• To multiply monomials, multiply the coefficients with their signs and add the powers of like variables.
• To multiply a term or expression by a bracket, distribute it across every term inside.
• After expanding, always collect like terms to give the simplest final answer.