1. Introduction
Multiplication of algebraic expressions uses the same idea as multiplying numbers, but with variables. When multiplying expressions, we multiply the coefficients (number parts) and then multiply the variables using the laws of exponents.
Understanding how to multiply expressions helps you simplify formulas, expand brackets, and solve algebraic equations with confidence.
2. Basic Rule of Multiplying Algebraic Expressions
Rule: Multiply the coefficients and then multiply the variables.
Example:
\( (3x)(2x) = 3 \times 2 \times x \times x = 6x^2 \)
This works because of the exponent rule: \( x^m x^n = x^{m+n} \).
2.1. Using Exponent Laws
When multiplying variables with the same base, add their powers.
- \(x^2 \times x^3 = x^{2+3} = x^5\)
- \(a \times a^4 = a^5\)
- \(mn \times m = m^2n\)
3. Multiplying a Monomial by a Monomial
To multiply two monomials, multiply their coefficients and their variables.
Example:
\((5x)(3x^2) = 15x^3\)
3.1. Worked Examples
- \((4a)(2a^2) = 8a^3\)
- \((-3m)(5m^3) = -15m^4\)
- \((2xy)(3x) = 6x^2y\)
4. Multiplying a Monomial by a Binomial
Use the distributive property: multiply the monomial with each term of the binomial.
Example:
\(3x(2x + 5) = 3x \times 2x + 3x \times 5 = 6x^2 + 15x\)
4.1. Worked Examples
Example 1:
\(4a(3a - 2) = 12a^2 - 8a\)
Example 2:
\((-2m)(5m + 7) = -10m^2 - 14m\)
5. Multiplying a Polynomial by a Monomial
Multiply the monomial by each term of the polynomial.
Example:
\(2x(3x + 5y - 7) = 6x^2 + 10xy - 14x\)
5.1. Worked Examples
- \(3m(4m^2 + m - 2) = 12m^3 + 3m^2 - 6m\)
- \(-5x(3x - 4y + 6) = -15x^2 + 20xy - 30x\)
6. Multiplying a Binomial by a Binomial
Use the distributive property or the FOIL method (First, Outer, Inner, Last).
Example:
\((a + b)(a + b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2\)
6.1. FOIL Method Explained
- First: multiply the first terms.
- Outer: multiply the outer terms.
- Inner: multiply the inner terms.
- Last: multiply the last terms.
6.1.1. Worked Examples
Example 1:
\((x + 3)(x + 2) = x^2 + 2x + 3x + 6 = x^2 + 5x + 6\)
Example 2:
\((2a - 5)(a + 4) = 2a^2 + 8a - 5a - 20 = 2a^2 + 3a - 20\)
7. Multiplying a Polynomial by a Polynomial
Multiply each term of the first polynomial with each term of the second polynomial.
Example:
\((x + 2)(x^2 + x + 1) = x^3 + x^2 + x + 2x^2 + 2x + 2 = x^3 + 3x^2 + 3x + 2\)
7.1. Worked Example
\((a + b + c)(a + b) = a(a + b) + b(a + b) + c(a + b) = a^2 + ab + ab + b^2 + ac + bc\)
8. Common Mistakes
- Not multiplying every term correctly.
- Forgetting exponent laws.
- Missing negative signs.
- Incorrect use of FOIL for expressions with more than two terms.
9. Quick Practice
Multiply and simplify:
- \(4x(3x - 2)\)
- \((x + 5)(x + 3)\)
- \((2a - 3)(a + 4)\)
- \((m + 2)(m^2 - m + 1)\)
10. Summary
- Multiply coefficients and then variables.
- Use exponent laws: add powers when multiplying same variables.
- Apply distributive property for binomials and polynomials.
- FOIL works only for binomial × binomial.
- Multiply carefully to avoid missing terms or signs.