Course

O Level Additional Mathematics: Polynomials and Equations (4037)

Name: O Level Additional Mathematics: Polynomials and Equations (4037)
Price: 999 BDT

Simultaneous equations, factors of polynomials, modulus equations and inequalities, cubic graphs and substitution for Additional Mathematics 4037.

45 lessons8h 25mEnglish

What you'll learn

Solve simultaneous equations by elimination or substitution, including one linear and one non-linear equation
Divide a polynomial by a linear expression and use the factor and remainder theorems
Factorise cubic polynomials and solve cubic equations
Solve modulus equations and inequalities by algebra and by graph
Sketch cubic graphs and their moduli from three linear factors, and solve cubic inequalities graphically
Use a substitution to turn a harder equation into a quadratic and solve it

Requirements

Cambridge O Level Mathematics 4024 or equivalent, including indices, solving quadratics and straight-line graphs
For the last substitution lesson only: eˣ and ln x from the sibling logarithms course

Course content

45 lessons · 8h 25m

5.1 · Simultaneous equations

Solve two equations in two unknowns by elimination or substitution, first with two straight lines, then with one linear and one non-linear equation, and pair every x with its own y.

0/9

Simultaneous equations: elimination and substitution

Two facts about two unknowns fix both. Add or subtract to remove one letter, or replace it with what the other equation says.

Preview12m 2.

Simultaneous equations: one linear and one non-linear

Make y the subject of the straight line and put it into the curve. The result is a quadratic in x with up to two answers.

Preview12m

Non-linear simultaneous equations with xy and squared terms

When the curve has xy, x² and y² together, substitute the linear equation into every term and expand each bracket with care.

13m

Non-linear simultaneous equations with products and fractions

Divide one product equation by the other, or clear the fractions first. A letter that ends up in a denominator can never be zero.

13m

Simultaneous equations: pairing and checking solutions

Each x has its own y from the linear equation. On the graph, each pair is one point where the line meets the curve.

11m

Simultaneous equations in context

Turn the words into two equations, solve them, then keep only the answers that make sense, such as a length that is positive.

13m

Simultaneous equations: practice

A timed mixed drill: linear pairs, a line with a curve, xy terms, products and a word problem.

15m

Simultaneous equations: exam practice

Paper-style questions with marks, full worked solutions and the mark scheme an examiner would use.

15m

Simultaneous equations: summary

The methods, the checks and the common mistakes of this chapter on one screen.

3.1-3.3 · Factors of polynomials

Work with polynomials, divide them by a linear expression, use the factor and remainder theorems, factorise a cubic into three factors and solve cubic equations.

0/12

Polynomials: adding, subtracting and multiplying

A polynomial is a sum of whole-number powers of x. Its degree is the highest power. Collect like terms, and multiply every term by every term.

Preview11m 2.

Division of polynomials

Long division works for polynomials as it does for numbers: divide the first terms, multiply back, subtract, bring down.

Preview13m

Division of polynomials: missing terms and remainders

Write 0x² for a missing power so the columns line up. What is left at the end is the remainder, and P(x) = divisor × quotient + remainder.

12m

The factor theorem

If P(a) = 0, then (x - a) is a factor of P(x). Test the numbers that divide the constant term instead of dividing each time.

12m

Using the factor theorem to find unknown coefficients

A known factor gives an equation P(a) = 0. One unknown needs one factor; two unknowns need two facts and a pair of simultaneous equations.

13m

Factorising cubic polynomials

Find one linear factor, divide or compare coefficients to get the quadratic factor, then factorise the quadratic if you can.

13m

Solving cubic equations

Factorise, then set each factor equal to zero. A quadratic factor that will not factorise needs the quadratic formula, and may give no real roots.

13m

The remainder theorem

The remainder when P(x) is divided by (x - a) is P(a). For (ax - b), substitute x = b/a.

12m

Using the remainder theorem to find unknown coefficients

Each remainder you are told gives one equation. Combine a remainder and a factor to find two unknowns, then use them.

13m

10.

Factors of polynomials: practice

A timed mixed drill: divide, test factors, find unknowns, factorise and solve cubics.

15m

11.

Factors of polynomials: exam practice

Paper-style questions, one with and one without a calculator, with full worked solutions and mark schemes.

15m

12.

Factors of polynomials: summary

Division, both theorems, factorising and solving cubics, and the common mistakes, on one screen.

4.1-4.2 · Equations, inequalities and graphs: the modulus

Read |x| as a distance from zero, draw y = |ax + b|, then solve every type of modulus equation and inequality in the syllabus, by algebra and by graph.

0/11

The modulus: solving |ax + b| = c

The modulus |u| is the distance of u from zero, so it is never negative. |u| = c means u = c or u = -c.

Preview11m 2.

Graphs of y = |ax + b|

Draw y = ax + b, then reflect the part below the x-axis upwards. The result is a V with its corner on the x-axis.

Preview12m

Solving |ax + b| = cx + d

Solve ax + b = cx + d and ax + b = -(cx + d), then reject any answer that makes the right side negative.

12m

Solving |ax + b| = |cx + d|

Two numbers with the same size are equal or opposite. Solve both cases, or square both sides and solve the quadratic.

13m

Solving |ax² + bx + c| = d

Solve the quadratic equal to d and equal to -d. The graph shows why there can be four, three, two or no answers.

13m

Modulus inequalities: k|ax + b| > c and k|ax + b| ≤ c

Less than a distance means one band between two values. Greater than a distance means two separate parts, outside both values.

12m

Modulus inequalities: |ax + b| ≤ |cx + d| and |ax + b| ≤ cx + d

Find where the two graphs meet, then read off where one lies below the other. Squaring both sides gives the same boundary values.

13m

Modulus inequalities: |ax² + bx + c| > d and |ax² + bx + c| ≤ d

Find where the folded parabola meets the line y = d, then read the x-values where the graph is above or below it.

13m

Modulus equations and inequalities: practice

A timed mixed drill of every modulus equation and inequality type in the syllabus.

15m

10.

Modulus equations and inequalities: exam practice

Paper-style questions that ask for a sketch and a solution, with full worked solutions and mark schemes.

15m

11.

Modulus equations and inequalities: summary

Every modulus method, the checks and the common mistakes, on one screen.

4.3-4.5 · Equations, inequalities and graphs: cubics and substitution

Sketch a cubic and its modulus from three linear factors, solve cubic inequalities from the graph, and spot the hidden quadratic inside harder equations.

0/9

Sketching cubic graphs from three linear factors

Each factor gives a root on the x-axis, x = 0 gives the y-intercept, and the sign of the x³ term fixes which way the ends go.

Preview12m 2.

Sketching cubic graphs with a repeated factor

A squared factor gives a root where the curve touches the x-axis and turns back, instead of crossing it.

Preview11m

Graphs of the modulus of a cubic

Sketch the cubic, then reflect every part below the x-axis upwards. The roots and the y-intercept size stay the same.

12m

Solving cubic inequalities graphically

Draw the cubic and the line y = d. Read the x-values where the curve is above or below the line, and use the right inequality signs.

13m

Using substitution to solve a related equation: powers of x

In x⁴ - 5x² + 4 = 0, let u = x². Solve the quadratic in u, then turn each u back into x and reject what is impossible.

13m

Using substitution to solve a related equation: exponentials and logarithms

Let u = 2ˣ, eˣ or ln x to reveal a quadratic. Remember that 2ˣ and eˣ are always positive, so a negative u gives no answer.

14m

Cubic graphs and substitution: practice

A timed mixed drill: sketch features, modulus cubics, cubic inequalities and hidden quadratics.

Cubic graphs and substitution: exam practice

Paper-style sketch, inequality and substitution questions with full worked solutions and mark schemes.

15m

Cubic graphs and substitution: summary

Sketch checklists, the inequality reading rule, the substitution steps and the common mistakes, on one screen.

About the creator

8bitlesson

Cambridge O Level, A Level and university courses, taught through interactive labs.

Mentora builds the sciences and mathematics as things you can move rather than videos you watch. Cambridge O Level and A Level mathematics, physics, chemistry and accounting, a primary mathematics track, and first-year university computer science. Every lesson works a real exam question end to end.