Products (Grade 10 NSC Matric Mathematics): Revision Notes

Products

Understanding algebraic expressions and their components

Before working with products in algebra, you need to understand the building blocks of algebraic expressions. Each part of an expression has a specific name and purpose.

Let's break down these key terms using a clear example. Consider the expression $3x^2 + 7xy - 5^3$:

• Term: Each separate part of the expression (like $3x^2$, $7xy$, or $-5^3$)
• Expression: The entire mathematical statement containing one or more terms
• Coefficient: The numerical part that multiplies a variable (3 and 7 in our example)
• Exponent: The small number that shows how many times the base is multiplied by itself (2, 1, and 3)
• Base: The number or variable being raised to a power (x, y, and 5)
• Constant: A number without a variable (like $-5^3 = -125$)
• Variable: Letters that represent unknown values (x and y)
• Equation: When an expression equals something (like $3x^2 + 7xy - 5^3 = 0$)

Types of algebraic expressions

Understanding the different types of expressions helps you choose the right multiplication method:

• Monomial: An expression with exactly one term
  • Examples: $3x$, $y^2$, $-5$, $4abc$
• Binomial: An expression with exactly two terms
  • Examples: $ax + b$, $cx + d$, $2x - 7$
• Trinomial: An expression with exactly three terms
  • Examples: $ax^2 + bx + c$, $x^2 - 2x + 1$

Multiplying a monomial and a binomial

When multiplying a single term by a two-term expression, use the distributive property. This means you multiply the monomial by each term in the binomial separately.

Method:

$a(b + c) = ab + ac$

Multiplying two binomials

When multiplying two binomial expressions, each term in the first binomial must be multiplied by each term in the second binomial.

General method:

$(axe + b)(cx + d) = axe(cx) + axe(d) + b(cx) + b(d)$

$= acx^2 + adx + bcx + bd$

$= acx^2 + x(ad + bc) + bd$

Special products

Perfect squares

When you multiply a binomial by itself, you get a perfect square:

Formula: $(axe + b)^2 = a^2x^2 + 2abx + b^2$

Pattern: Square the first term, twice the product of both terms, square the second term

Difference of squares

When you multiply two binomials that differ only in the sign between terms:

Formula: $(axe + b)(axe - b) = a^2x^2 - b^2$

Pattern: This always gives the difference between the squares of the two terms

Multiplying a binomial and a trinomial

To multiply a binomial by a trinomial, multiply each term in the binomial by every term in the trinomial.

Method:

$(A + B)(C + D + E) = A(C + D + E) + B(C + D + E)$

Exam tips for products

Common exam traps