Algebra Revision Notes

Multiplying Algebraic Terms

When multiplying algebraic terms, remember two key steps. First, multiply the numerical coefficients. These are the numbers in front of the variables. For example, in $2x \times 3y$, the coefficients are 2 and 3. Second, multiply the variables. If the variables are the same, add their powers. If they are different, write them next to each other. For instance, $2x \times 3x^2 = (2 \times 3) \times (x \times x^2) = 6x^{(1+2)} = 6x^3$. For different variables like $4a \times 5b$, the result is $20ab$.

Dividing Algebraic Terms

When dividing algebraic terms, follow a similar structure to multiplication. First, divide the numerical coefficients. For example, in $\frac{10y^4}{2y}$, divide 10 by 2. Second, divide the variables. If the variables are the same, subtract their powers. Remember that a variable without an explicit power has a power of 1. So, $\frac{10y^4}{2y} = (10 \div 2) \times (y^4 \div y^1) = 5y^{(4-1)} = 5y^3$. If terms don't cancel completely, leave them as fractions, e.g., $\frac{7x}{2y}$ cannot be simplified further.

Combining Operations

When an expression involves both multiplication and division, perform the operations step-by-step. It's often helpful to simplify the numerator and denominator separately before performing the division. For example, to simplify $\frac{4a^2b \times 3ab^3}{6a^3}$, first simplify the numerator: $4a^2b \times 3ab^3 = 12a^3b^4$. Then perform the division: $\frac{12a^3b^4}{6a^3} = (12 \div 6)a^{(3-3)}b^4 = 2a^0b^4$. Since $a^0 = 1$, the final simplified form is $2b^4$. Pay close attention to each variable and its power.

Multiplying Algebraic Terms

When multiplying algebraic terms, multiply the numerical coefficients together and combine the variables. For variables with the same base, add their powers according to the index law $a^m \times a^n = a^{m+n}$. For example, to simplify $5x^3y^2 \times 2xy^4$: multiply the coefficients ($5 \times 2 = 10$), then combine $x$ terms ($x^3 \times x = x^{3+1} = x^4$), and combine $y$ terms ($y^2 \times y^4 = y^{2+4} = y^6$). The simplified product is $10x^4y^6$. Remember that a variable written without an explicit power has a power of 1, e.g., $x = x^1$.

Dividing Algebraic Terms

When dividing algebraic terms, divide the numerical coefficients and subtract the powers of variables with the same base according to the index law $a^m \div a^n = a^{m-n}$. For example, to simplify $\frac{18p^5q^3}{3p^2q}$: divide the coefficients ($18 \div 3 = 6$), then combine $p$ terms ($p^5 \div p^2 = p^{5-2} = p^3$), and combine $q$ terms ($q^3 \div q = q^{3-1} = q^2$). The simplified quotient is $6p^3q^2$. If the power of a variable in the denominator is greater than in the numerator, the result will be a negative power, e.g., $x^2 \div x^5 = x^{-3} = \frac{1}{x^3}$.

Simplifying Rational Expressions by Factorising

For algebraic fractions (rational expressions), simplification often involves factorising both the numerator and the denominator, and then cancelling out any common factors. This process relies on strong factorisation skills, including common factorisation, difference of two squares, and factorising quadratic expressions. For example, to simplify $\frac{3x^2 - 12}{x^2 - 4x + 4}$: First, factorise the numerator $3x^2 - 12 = 3(x^2 - 4) = 3(x-2)(x+2)$. Next, factorise the denominator $x^2 - 4x + 4 = (x-2)(x-2)$. So, the expression becomes $\frac{3(x-2)(x+2)}{(x-2)(x-2)}$. Now, cancel the common factor $(x-2)$ from both the numerator and the denominator. The simplified expression is $\frac{3(x+2)}{x-2}$. This method applies to expressions where common factors are binomials, not just single terms.

The Distributive Law

Multiplying out brackets relies on the distributive law: $a(b+c) = ab + ac$. This means whatever is outside the bracket multiplies by EACH term inside the bracket. It's crucial to apply this to all terms, not just the first. For example, $4(x+y) = 4x+4y$. Similarly, $x(y+z) = xy+xz$. If the term outside is a negative number, like $-2(3x-5)$, multiply $-2$ by $3x$ to get $-6x$, and $-2$ by $-5$ to get $+10$, resulting in $-6x+10$. Be careful with signs: a negative multiplied by a negative gives a positive (+).

Multiplying a Single Bracket

To multiply a single bracket by a term, distribute the outside term to every term inside. For example, to expand $5(2x+3)$: multiply $5 \times 2x = 10x$, and $5 \times 3 = 15$. The expanded form is $10x+15$. Another example: $x(x-7)$. Multiply $x \times x = x^2$, and $x \times (-7) = -7x$. The result is $x^2-7x$. Always simplify the resulting expression by collecting like terms if possible, though often for single brackets, the terms are not 'like' after expansion.

Multiplying Two Brackets (Binomials)

When multiplying two brackets, such as $(a+b)(c+d)$, every term in the first bracket must be multiplied by every term in the second bracket. This can be remembered using the FOIL method: First, Outer, Inner, Last. For example, to expand $(x+2)(x-5)$:

1. First terms: $x \times x = x^2$
2. Outer terms: $x \times (-5) = -5x$
3. Inner terms: $2 \times x = 2x$
4. Last terms: $2 \times (-5) = -10$

Add these results: $x^2 - 5x + 2x - 10$. Finally, combine like terms: $x^2 - 3x - 10$. This method ensures all products are found. Remember to be meticulous with positive and negative signs during each multiplication step.

Understanding the Distributive Property

The core principle behind multiplying out brackets is the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. This means you 'distribute' the multiplication of the term outside the bracket to each term inside. For example, 3(x + 5) means 3 multiplied by x, PLUS 3 multiplied by 5, resulting in 3x + 15. Always ensure every term inside the bracket is multiplied by the term outside.

Multiplying with Negative Signs

When multiplying out brackets, be very careful with negative signs. If the term outside the bracket is negative, or if a term inside the bracket is negative, the rules for multiplying positive and negative numbers apply. For instance, -4(y + 2) becomes (-4) * y + (-4) * 2, which simplifies to -4y - 8. Another example: 5(2 - x) becomes 5 * 2 + 5 * (-x), which simplifies to 10 - 5x. Remember: negative times negative equals positive, negative times positive equals negative.

Combining Like Terms After Expansion

After multiplying out brackets, you often need to simplify the expression further by collecting like terms. Like terms are terms that have the same variable raised to the same power (e.g., 3x and 7x, or 2y² and -5y²). You can add or subtract the coefficients of like terms. For example, expand and simplify 2(x + 3) + 4x. First, expand: 2x + 6 + 4x. Then, collect like terms: (2x + 4x) + 6 = 6x + 6. Make sure you can identify which terms can be combined.