Algebra and Functions (Edexcel IAL P1)

Q: Do I need to show full working for surd rationalisation questions?

Yes, Edexcel awards explicit method marks for all algebraic steps when rationalising denominators, even if your final answer is correct. Skipping steps can cost you marks if you make an arithmetic error.

Q: What is the highest degree of polynomial I need to factorise for P1?

Only polynomials of degree ≤ 3 (linear, quadratic, cubic) are tested in P1. Rational expression simplification and algebraic long division are out of scope for this unit.

Q: Do I need to label asymptotes when sketching reciprocal graphs?

Yes, you must explicitly draw and label the x and y asymptotes for reciprocal graphs to get full marks for sketching questions in P1 exams.

Mathematics· 2018 Edexcel IAL Maths P1 (WMA11) Specification sections 1.1-1.12· 45 min read

1. Indices and Surds★★☆☆☆⏱ 10 min

✓ Calculator OK

📘 Definition

Laws of Indices

For all rational exponents $m, n$ and non-zero base $a$ : $a^{m} \times a^{n} = a^{m + n}$ , $a^{m} \div a^{n} = a^{m - n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{0} = 1$ , $a^{- n} = 1/ a^{n}$ , $a^{1/ n} = n a$

📐 Worked Example

Simplify $\frac{( 16 x ^{4} y ^{- 2} ) ^{1/2}}{8 x ^{3} y}$

1
Step 1: Apply power rule to the numerator
$(16x^4 y^{-2})^{1/2} = 16^{1/2} (x^4)^{1/2} (y^{-2})^{1/2} = 4x^2 y^{-1}$
2
Step 2: Divide by the denominator, subtract exponents for matching bases
$\frac{4x^2 y^{-1}}{8x^3 y} = 0.5 x^{2-3} y^{-1-1} = 0.5 x^{-1} y^{-2} = \frac{1}{2xy^2}$

📘 Definition

Surd Manipulation

Surds follow the rules: $ab = a \times b$ and $a / b = a / b$ . Rationalising the denominator removes all surds from the bottom of a fraction, using the conjugate pair for denominators with two terms.

📐 Worked Example

Rationalise the denominator of $\frac{3}{2 + 3}$

1
Step 1: Multiply numerator and denominator by the conjugate of the denominator, $2 - 3$
$\frac{3(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}$
2
Step 2: Simplify the denominator using difference of squares
$(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1$
3
Step 3: Final simplified answer
$6 - 3\sqrt{3}$

Exam tip:

Always leave surd answers in fully simplified form, with no surds in denominators, unless the question explicitly states otherwise.

2. Quadratic Functions and Equations★★★☆☆⏱ 12 min

✓ Calculator OK

📘 Definition

Quadratic Function Properties

A quadratic function has the form $f (x) = a x^{2} + b x + c$ , $a \neq = 0$ . Its graph is a parabola opening upwards if $a > 0$ and downwards if $a < 0$ . The discriminant $Δ = b^{2} - 4 a c$ gives the number of real roots: $Δ > 0$ = 2 distinct real roots, $Δ = 0$ = 1 repeated real root, $Δ < 0$ = no real roots.

📐 Worked Example

Find the value of $k$ for which the equation $2 x^{2} + k x + 8 = 0$ has exactly one repeated real root.

1
Step 1: Identify coefficients $a = 2$ , $b = k$ , $c = 8$ . For a repeated root, $Δ = 0$
$b^2 - 4ac = 0$
2
Step 2: Substitute values and solve for $k$
$k^2 - (4 \times 2 \times 8) = 0 \implies k^2 = 64 \implies k = \pm 8$

📘 Definition

Completing the Square

Rewrites $a x^{2} + b x + c$ as $a (x + p)^{2} + q$ , where $p = b / (2 a)$ and $q = c - b^{2} / (4 a)$ . This form gives the vertex of the parabola at $(- p, q)$ .

📐 Worked Example

Solve $x^{2} - 4 x - 5 = 0$ by completing the square.

1
Step 1: Group x terms
$(x^2 - 4x) - 5 = 0$
2
Step 2: Complete the square inside the bracket
$(x - 2)^2 - 4 - 5 = 0 \implies (x - 2)^2 = 9$
3
Step 3: Take square roots and solve
$x - 2 = \pm 3 \implies x = 5 \text{ or } x = -1$

Exam tip:

Always write the quadratic formula explicitly before substituting values to get method marks even if you make an arithmetic error during calculation.

3. Simultaneous Equations and Inequalities★★★☆☆⏱ 10 min

✓ Calculator OK

For simultaneous equations with one linear and one quadratic, use the substitution method: rearrange the linear equation to express one variable in terms of the other, substitute into the quadratic equation, solve, then substitute back to find the second variable.

📐 Worked Example

Solve the simultaneous equations: $y = 2 x + 1$ , $y = x^{2} + 3 x - 1$

1
Step 1: Substitute $y = 2 x + 1$ into the quadratic equation
$2x + 1 = x^2 + 3x - 1$
2
Step 2: Rearrange to standard quadratic form
$x^2 + x - 2 = 0$
3
Step 3: Factorise and solve for x
$(x + 2)(x -1) = 0 \implies x = -2, x = 1$
4
Step 4: Substitute x values back into linear equation to find y
$x=-2 \implies y=-3, x=1 \implies y=3. \\ \text{Solutions: } (-2, -3) \text{ and } (1, 3)$

For quadratic inequalities, always sketch the parabola first to identify the region that satisfies the inequality. For $a x^{2} + b x + c > 0$ , if $a > 0$ the solution is the regions outside the roots; if $a < 0$ it is the region between the roots.

📐 Worked Example

Solve $2 x^{2} - 5 x - 3 < 0$ and show the solution on a number line.

1
Step 1: Factorise the quadratic
$(2x +1)(x - 3) = 0 \implies \text{roots at } x = -0.5, x = 3$
2
Step 2: $a = 2 > 0$ so parabola opens upwards, so the region <0 is between the roots
$-0.5 < x < 3$
3
Step 3: On a number line, mark open circles at -0.5 and 3, and shade the region between them.

Exam tip:

When solving inequalities, never multiply or divide both sides by a negative number without reversing the direction of the inequality sign.

4. Polynomials and Core Graphs★★★☆☆⏱ 8 min

✓ Calculator OK

For P1, you only need to factorise polynomials of degree ≤ 3. Use the factor theorem: if $f (a) = 0$ , then $(x - a)$ is a factor of $f (x)$ .

📐 Worked Example

Factorise fully $f (x) = x^{3} - 4 x^{2} + x + 6$

1
Step 1: Test integer values of x to find a root. $f (2) = 8 - 16 + 2 + 6 = 0$ , so $(x - 2)$ is a factor
2
Step 2: Expand $(x - 2) (x^{2} + a x + b)$ and equate coefficients
$x^3 + (a -2)x^2 + (b - 2a)x - 2b = x^3 -4x^2 +x +6$
3
Step 3: Solve for a and b, then factorise the quadratic
$a=-2, b=-3, \text{ so } x^2 -2x -3 = (x-3)(x+1) \\ \text{Full factorisation: } (x-2)(x-3)(x+1)$

📘 Definition

Core Graphs for P1

Cubic graphs: $y = (x - a) (x - b) (x - c)$ have x-intercepts at $a, b, c$ , y-intercept at $- ab c$ . 2. Reciprocal graphs: $y = k / x$ has asymptotes at $x = 0$ and $y = 0$ , two curves in opposite quadrants. $y = k / x^{2}$ has the same asymptotes, two curves in the top two quadrants if $k > 0$ .

📐 Worked Example

Sketch the graph of $y = 4/ x^{2}$ , label all key features.

1
Step 1: Identify asymptotes: vertical asymptote $x = 0$ (y-axis), horizontal asymptote $y = 0$ (x-axis)
2
Step 2: $k = 4 > 0$ , so the graph has two U-shaped curves in the first and second quadrants, getting closer to the axes as x becomes very large or very small
3
Step 3: Label intercepts: there are no x or y intercepts, as the graph never touches the axes

Exam tip:

For all sketch questions, always label x-intercepts, y-intercepts and asymptotes (where applicable) to get full marks.

5. Graph Transformations★★★★☆⏱ 7 min

✓ Calculator OK

Transformation	Effect on graph of $f (x)$
$y = f (x) + a$	Shift vertically up by $a$ units
$y = f (x + a)$	Shift horizontally left by $a$ units
$y = a f (x)$	Stretch vertically by scale factor $a$ , parallel to y-axis
$y = f (a x)$	Stretch horizontally by scale factor $1/ a$ , parallel to x-axis

📐 Worked Example

The graph of $f (x) = x^{2}$ is transformed to give $g (x) = f (x) - 4$ . Describe the single transformation fully, and state the coordinates of the new turning point.

1
Step 1: $g (x) = f (x) - 4$ has the form $f (x) + a$ with $a = - 4$ .
2
Step 2: This is a single translation of 4 units down (parallel to the $y$ -axis).
3
Step 3: The turning point of $f (x) = x^{2}$ is $(0, 0)$ , so the new turning point is $(0, - 4)$ .

Exam tip:

When describing transformations, always state the direction, scale factor and axis parallel to which the stretch/shift occurs to get full marks.

6. Common Pitfalls

Wrong move:

Adding exponents instead of multiplying when raising a power to a power, e.g. writing $(x^{2})^{3} = x^{5}$

Why:

Confuses the product rule ( $a^{m} \times a^{n} = a^{m + n}$ ) with the power rule ( $(a^{m})^{n} = a^{mn}$ )

Correct move:

Multiply exponents when raising a power to a power, add exponents only when multiplying terms with the same base.

Wrong move:

Rationalising $1/ (2 + 3)$ by multiplying only by $3$ instead of the conjugate pair

Why:

Forgets that denominators with two terms require multiplying by the conjugate to eliminate all surds from the denominator

Correct move:

Multiply numerator and denominator by the conjugate of the denominator ( $2 - 3$ ) to get $2 - 3$ .

Wrong move:

Using discriminant on a quadratic that is not rearranged to equal 0, leading to incorrect root count

Why:

Applies the discriminant rule to unbalanced equations with terms on both sides

Correct move:

Always rearrange all quadratic terms to one side so the equation equals 0 before identifying $a, b, c$ for discriminant calculation.

Wrong move:

Giving solution $x < - 1$ or $x > 4$ for the inequality $x^{2} - 3 x - 4 < 0$

Why:

Fails to sketch the upward opening parabola, so mixes up the region where the graph is below the x-axis

Correct move:

Sketch the parabola first; for an upward opening parabola, values less than 0 lie between the roots, so solution is $- 1 < x < 4$ .

Wrong move:

Translating $f (x + 2)$ 2 units to the right instead of left

Why:

Forgets that x-axis transformations have the opposite effect to the sign inside the function bracket

Correct move:

Transformations inside the function bracket affect the x-axis in the opposite direction: $f (x + a)$ shifts the graph $a$ units left.

Wrong move:

Canceling terms when factorising a cubic as part of a rational expression

Why:

Rational expression simplification is out of scope for P1, and not tested in this unit

Correct move:

Only fully factorise the cubic into linear/quadratic factors, do not cancel any terms across a fraction line.

7. Quick Reference Cheatsheet

Topic	Key Rule	Exam Reminder
Indices	$a^{m} \times a^{n} = a^{m + n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{- n} = 1/ a^{n}$	Simplify exponents to positive rational or integer form
Surds	Rationalise denominators with the conjugate pair for 2-term denominators	No surds allowed in final answer denominators
Discriminant	$Δ = b^{2} - 4 a c$ : >0 = 2 roots, =0 = 1 root, <0 = no real roots	Rearrange quadratic to =0 first before calculating $Δ$
Completing the Square	$a x^{2} + b x + c = a (x + b / (2 a))^{2} + (c - b^{2} / (4 a))$	Gives the vertex of the parabola at $(- b / (2 a), c - b^{2} / (4 a))$
Simultaneous Equations	Substitute linear expression into quadratic, solve then substitute back	Check both solutions work for both original equations
Graph Transformations	Inside bracket = x-axis (opposite direction), outside = y-axis (normal direction)	Always state scale factor and axis for stretches

8. Frequently Asked

Do I need to show full working for surd rationalisation questions?

Yes, Edexcel awards explicit method marks for all algebraic steps when rationalising denominators, even if your final answer is correct. Skipping steps can cost you marks if you make an arithmetic error.

What is the highest degree of polynomial I need to factorise for P1?

Only polynomials of degree ≤ 3 (linear, quadratic, cubic) are tested in P1. Rational expression simplification and algebraic long division are out of scope for this unit.

Do I need to label asymptotes when sketching reciprocal graphs?

Yes, you must explicitly draw and label the x and y asymptotes for reciprocal graphs to get full marks for sketching questions in P1 exams.

Going deeper

official_specEdexcel IAL Maths 2018 SpecificationRefer to section 2.1 for P1 algebra content
practice_packP1 Algebra and Functions Practice QuestionsExam-style questions aligned to WMA11 specification

What's Next

Algebra and functions is the foundational topic for all other P1 content, so mastering these rules will make coordinate geometry, differentiation and integration significantly easier to learn. Make sure you practice 5-10 past paper questions on this topic to build speed and avoid common errors before moving to more advanced P1 content. The included cheatsheet can be used for quick last-minute revision before your exam. This content also underpins many topics in P2 and P3, so strong foundational knowledge will reduce your workload later in the course.

→Differentiation (Edexcel IAL P1)