Multiplication of Monomials

Definition: A monomial is an algebraic expression that contains only one term. It is a product of a constant (coefficient) and one or more variables raised to non-negative integer powers.

Parts of a monomial:

• Coefficient — The numerical part. In 7x²y, the coefficient is 7.
• Variables — The letters representing unknown values. In 7x²y, the variables are x and y.
• Exponents — The powers of the variables. In 7x²y, the exponent of x is 2 and of y is 1.
• Degree — The sum of all exponents. Degree of 7x²y = 2 + 1 = 3.

Examples of monomials:

• 5 (a constant is a monomial of degree 0)
• −3x (degree 1)
• 4x²y³ (degree 5)
• abc (degree 3, coefficient = 1)

NOT monomials:

• x + y (two terms — this is a binomial)
• 3x² − 5x + 2 (three terms — this is a trinomial)

Why do we add exponents when multiplying?

Consider x³ × x²:

• x³ = x × x × x (three factors of x)
• x² = x × x (two factors of x)
• x³ × x² = (x × x × x) × (x × x) = x × x × x × x × x = x⁵
• So x³ × x² = x³⁺² = x⁵.

The exponent counts how many times the base is used as a factor. When multiplying, we combine all factors, so the total count (exponent) is the sum.

Extending to multiple variables:

Consider 3x²y × 5xy³:

• Coefficients: 3 × 5 = 15
• Variable x: x² × x¹ = x²⁺¹ = x³
• Variable y: y¹ × y³ = y¹⁺³ = y⁴
• Result: 15x³y⁴

Multiplying three or more monomials:

The rule extends naturally. Multiply all coefficients together, and for each variable, add up all its exponents across all the monomials.

Monomial multiplication problems can be classified as follows:

1. Multiplying two simple monomials (one variable):

• Example: 3x⁴ × 7x² = 21x⁶

2. Multiplying monomials with multiple variables:

• Example: 2xy × 5x²y³ = 10x³y⁴

3. Multiplying monomials with negative coefficients:

• Example: (−4x³) × (3x²) = −12x⁵
• Sign rules: (+)(+) = +, (+)(−) = −, (−)(+) = −, (−)(−) = +

4. Multiplying three or more monomials:

• Example: 2x × 3y × 4z = 24xyz

5. Multiplying a monomial by a constant:

• Example: 5 × 3x²y = 15x²y

6. Multiplying monomials and finding the degree:

• If monomial A has degree m and monomial B has degree n, their product has degree m + n.

Finding Areas and Volumes:

• When dimensions of geometric shapes are given as monomials, their area or volume is found by multiplying monomials. For example, the area of a rectangle with sides 3x and 5y is 15xy.

Simplifying Algebraic Expressions:

• Many algebraic simplifications require multiplying monomials as an intermediate step. For example, expanding 3x(2x + 5) requires multiplying 3x by each term inside the bracket.

Physics Formulas:

• Physical quantities often involve products of variables with exponents. Force = mass × acceleration involves a product similar to monomial multiplication when variables have units with powers.

Building Toward Polynomial Multiplication:

• Multiplying binomials using the distributive property or FOIL method is essentially repeated monomial multiplication. Each pair of terms produces a monomial product.

Factorisation:

• Understanding how monomials multiply helps reverse the process — breaking an expression into its monomial factors.