8. Algebraic Expressions and Identities

In earlier classes, we have already become familiar with what algebraic expressions (or simply expressions) are. Examples of expressions are: x + 3, 2y – 5, 3x², 4xy + 7 etc.

Algebraic expressions involve variables, constants, and operations. A variable is a symbol (like x or y) that can represent a number. A constant is a fixed number (like 3 or 5). An expression combines these elements using operations like addition, subtraction, and multiplication. The examples provided help illustrate different forms of these expressions: single variables, terms with coefficients, and combinations of variables and constants together.

Think of algebraic expressions like shopping lists. If 'x' represents the number of apples and 'y' represents oranges, 'x + 3' could mean buying 'x' apples and 3 bananas. Just like changing the number of items changes your expense, changing variables in expressions alters their value.

For example, to add 7x² – 4x + 5 and 9x – 10, we do: 7x² – 4x + 5

• 9x – 10 = 7x² + 5x – 5.

To add algebraic expressions, align like terms vertically. Like terms share the same variable part; for instance, the terms '–4x' and '9x' are like terms. You can add the coefficients directly: –4 + 9 = 5, yielding '5x'. The process is similar for constants: 5 + (–10) results in –5. The final expression combines all simplified terms.

Imagine combining two different fruit baskets. If one basket has '7 apples', '4 bananas', and '5 oranges', and the other has '9 bananas' and '10 oranges', you would add your apples separately from your bananas and oranges to see what you have in total.

Solution: 7x² – 4xy + 8y² + 5x – 3y – (5x² – 4y² + 6y – 3) = 2x² – 4xy + 12y² + 5x – 9y + 3.

Subtraction in algebraic expressions also involves aligning like terms and carefully managing signs. Each term from the second expression (the one being subtracted) needs its sign changed according to the subtraction operation. For example, subtracting +5x² switches it to –5x². This change must then be combined with the first expression's terms. Collect similar terms to form the final expression.

Imagine you have a certain number of fruits, and you give some away. If you have '7 bananas' but give away '5 bananas', to find out how many bananas you have left, visualize '7' and '–5' being added, which results in '2 bananas'. The signs represent whether you have or lose items.

We shall look at the multiplication of two monomials. An expression that contains only one term is called a monomial.

Example: 4 × (3x) = 12x.

Multiplying monomials involves simple arithmetic on their coefficients and applying the rules for exponents on variable parts. For example, multiplying '4' (a coefficient) by '3x' (a monomial) gives '12x'. Similarly, x multiplied by x (x^1 * x^1) becomes x² due to the exponent rule which states to add exponents.

Think of monomials as individual fruit trees. If one tree produces '4 fruits' and you have '3 trees', the total fruit is like multiplying, resulting in '12 fruits'. Each tree contributes to the total yield just like each term contributes in an expression.

Expression that contains two terms is called a binomial. Multiplying a monomial by a binomial, e.g., 3x × (5y + 2) results in 15xy + 6x.

When multiplying a monomial by a binomial, distribute the monomial to each term in the binomial. For 3x × (5y + 2), you multiply 3x by both 5y and 2 individually. The process involves adhering to the distributive property, ensuring that each multiplication occurs which helps combine terms efficiently.

Consider a scenario where you pay for two different items in a shop. If you spend '3 dollars' on '5 apples' and '2 oranges', you would calculate your total cost separately by multiplying your purchase amount with quantities. The way your money is distributed relates closely to how we distribute terms in an expression.

In multiplication of polynomials with polynomials, we should always look for like terms if any, and combine them.

Example: (2a + 3b) × (3a + 4b) = 6a² + 17ab + 12b².

Multiply each term in one polynomial by every term in the other polynomial. This involves careful multiplication, followed by the need to combine like terms. For instance, in the product of polynomials (2a + 3b) and (3a + 4b), you multiply 2a with both 3a and 4b, and 3b with both, and finally sum all resulting terms while combining any like terms.

If you have two types of plants (let’s say '2 types of apple trees' and '3 types of orange trees') and each type can produce different varieties, the combinations of each type teaching show how many varieties can be created, reflecting how polynomial terms interact, leading to a diverse range of results.

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