Welcome to the World of Algebra!
Welcome to your study notes for P1: Pure Maths - Algebra. Think of Algebra not just as a series of rules, but as a mathematical "toolkit." Once you master these tools, you’ll be able to unlock much more complex problems in calculus, geometry, and beyond. This chapter covers everything from "cleaning up" numbers (surds) to solving equations and understanding the shapes of graphs.
Don't worry if some of this seems tricky at first—maths is a skill that grows with practice. Let's dive in!
1. Surds and Indices: Mastering the Basics
Before we build the house, we need to understand the bricks. Surds and indices are the building blocks of algebraic expressions.
Surds
A surd is an irrational number that is left in its root form (like \(\sqrt{2}\) or \(\sqrt{3}\)). We use them because they are exact. For example, \(1.41\) is just an estimate, but \(\sqrt{2}\) is perfectly precise!
Key Rules to Remember:
1. \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
2. \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
Rationalising the Denominator:
In maths, we don't like having roots on the bottom of a fraction. It’s like wearing your socks over your shoes—it’s just messy! To "fix" it, we multiply the top and bottom by a value that cancels the root.
Example: To rationalise \(\frac{1}{\sqrt{2} - 1}\), we multiply the top and bottom by \((\sqrt{2} + 1)\). This uses the "difference of two squares" to remove the root from the bottom.
Indices (Powers)
Indices tell us how many times to multiply a number by itself. You need to be comfortable with rational exponents (powers that are fractions).
Quick Review:
• \(x^a \times x^b = x^{a+b}\)
• \(x^a \div x^b = x^{a-b}\)
• \((x^a)^b = x^{ab}\)
• \(x^{\frac{1}{n}} = \sqrt[n]{x}\) (The bottom of the fraction is the root!)
Common Mistake to Avoid:
Remember that \(x^0 = 1\), not \(0\)! Also, a negative power like \(x^{-2}\) means "one over," so \(x^{-2} = \frac{1}{x^2}\). It does not make the number negative.
Key Takeaway: Surds keep your answers exact. Indices follow strict rules—learn the rules, and the problems solve themselves!
2. Quadratic Functions
A quadratic is any expression where the highest power of \(x\) is \(x^2\). They always draw a U-shaped or n-shaped curve called a parabola.
The Three Ways to Solve Quadratics
When an exam asks you to "solve" \(ax^2 + bx + c = 0\), you have three main tools:
1. Factorisation: Putting the expression into two brackets. Example: \(x^2 + x - 6 = (x+3)(x-2)\).
2. The Quadratic Formula: The "old reliable" method. Use this when factorising is too hard: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
3. Completing the Square: This involves writing the quadratic in the form \((x+p)^2 + q\). This is very useful for finding the vertex (the turning point) of the graph.
The Discriminant: The "Mood Meter"
The part of the formula under the root, \(b^2 - 4ac\), is called the discriminant. It tells us how many times the graph hits the x-axis:
• If \(b^2 - 4ac > 0\): Two distinct real roots (The graph hits the axis twice).
• If \(b^2 - 4ac = 0\): One real root (Equal roots; the graph just touches the axis at the vertex).
• If \(b^2 - 4ac < 0\): No real roots (The graph is floating above or below the axis).
Did you know? The line of symmetry of a quadratic graph always passes through the vertex. If you’ve completed the square to get \((x-3)^2 + 5\), the line of symmetry is simply \(x = 3\).
Key Takeaway: The discriminant is a shortcut. It tells you the nature of the roots without you having to solve the whole equation!
3. Simultaneous Equations and Inequalities
Sometimes you have to deal with two equations at once, or equations that aren't "equals" but are "greater than" or "less than."
Simultaneous Equations
For P1, you will often have one linear equation (like \(y = x + 2\)) and one quadratic equation (like \(y = x^2 - 4\)).
Step-by-Step Process:
1. Rearrange the linear equation to get \(y = ...\) or \(x = ...\).
2. Substitute this into the quadratic equation.
3. Solve the resulting quadratic for one variable.
4. Plug those answers back into the linear equation to find the other variable.
Quadratic Inequalities
Solving \(x^2 + x - 6 > 0\) is different from solving an equation.
The "Sketch Method":
1. Find the "critical values" by solving the equation as if it were \(= 0\).
2. Sketch the parabola.
3. If the question asks for \(> 0\), look for the parts of the graph above the x-axis. If \(< 0\), look for the parts below.
Key Takeaway: For inequalities, always draw a sketch! It’s the easiest way to make sure you don't get your "greater than" and "less than" signs mixed up.
4. Polynomials: Division and Theorems
A polynomial is just an expression with many terms, like a cubic: \(ax^3 + bx^2 + cx + d\).
Algebraic Division
You can divide a polynomial by a linear term like \((x - a)\). This is just like long division you did in primary school, but with letters! You can also do this "by inspection" by matching up the coefficients.
The Factor and Remainder Theorems
These two theorems are huge time-savers.
• The Remainder Theorem: If you divide a polynomial \(f(x)\) by \((x - a)\), the remainder is simply \(f(a)\).
• The Factor Theorem: If you plug \(a\) into the function and get zero (\(f(a) = 0\)), then \((x - a)\) is a factor of the polynomial.
Memory Aid: "Zero is the Hero"
If the result is zero, you’ve found a factor! This is how we break down big cubic equations into smaller, solvable parts.
Example: If \(f(x) = x^3 - 5x^2 + 7x - 3\) and you find that \(f(1) = 0\), then you know for sure that \((x - 1)\) is a factor.
Key Takeaway: Before you start long division, use the Factor Theorem to check if a bracket is actually a factor. It will save you a lot of work!
5. Graphs and Intersections
Algebra and Geometry are two sides of the same coin. An equation is just a rule that creates a shape on a graph.
Sketching Curves
You should be able to recognize and sketch:
• Linear: A straight line.
• Quadratic: A U-shaped or n-shaped parabola.
• Cubic: An S-shaped curve (usually has two turning points).
Intersections
If you have two functions, \(f(x)\) and \(g(x)\), the points where their graphs cross are the solutions to the equation \(f(x) = g(x)\).
• 2 intersection points = 2 real solutions.
• 1 intersection point (tangent) = 1 repeated solution.
• No intersection = No real solutions.
Quick Review Box:
• Vertex: The peak or valley of a quadratic.
• Roots: Where the graph crosses the x-axis.
• y-intercept: Where the graph crosses the y-axis (set \(x = 0\)).
Key Takeaway: If you're stuck on an algebraic problem, try to visualize what the graphs are doing. Often, the "geometry" makes the "algebra" much clearer!