Course

Cambridge A Level Further Mathematics: Further Pure 1 (9231 Paper 1)

Name: Cambridge A Level Further Mathematics: Further Pure 1 (9231 Paper 1)
Price: 1499 BDT

Paper 1 of 9231, Further Pure Mathematics 1. Written so far: topic 1.1, roots of polynomial equations, taught step by step from quadratics to quartics, with practice, exam questions and a summary.

16 lessons3h 5mEnglish

What you'll learn

Use the words degree, coefficient and root, and name the roots of an equation α, β, γ and δ
Write down the sum, the sum of products in pairs (and threes) and the product of the roots of a quadratic, cubic or quartic
Form a quadratic equation with integer coefficients from its two roots
Find symmetric functions of the roots, such as α² + β² or Σ1/α, without solving the equation
Find sums of powers of the roots using the equation itself
Find unknown coefficients when the roots are linked in a given way
Use a substitution to form the equation whose roots are α + k, kα, 1/α or α², and use that equation to find new values

Requirements

Pure Mathematics 1 of 9709, from our course or from school: quadratics and the factor theorem
Confident expanding brackets and working with fractions in algebra

Course content

16 lessons · 3h 5m

1.1 · Roots of polynomial equations

Read the sum, pairs and product of the roots from the coefficients of a quadratic, cubic or quartic. Use them to find symmetric functions, sums of powers and unknown coefficients, and form new equations by substitution.

0/15

Polynomials and their roots

What degree, coefficient and root mean, why a polynomial of degree n has n roots, and how we name them α, β, γ, δ.

Preview10m 2.

Roots and coefficients of a quadratic equation

Multiply out a(x − α)(x − β) and compare: the sum of the roots is −b/a and the product is c/a.

Preview12m

Forming a quadratic equation from its roots

Any quadratic with roots α and β can be written x² − (sum)x + (product) = 0, then cleared of fractions.

10m

Roots and coefficients of a cubic equation

Three roots give three facts: Σα = −b/a, Σαβ = c/a and αβγ = −d/a, with signs that take turns.

12m

Roots and coefficients of a quartic equation

Four roots give four facts, Σα, Σαβ, Σαβγ and αβγδ, and the same alternating signs.

12m

Symmetric functions of the roots of a quadratic

Swapping α and β leaves these unchanged, so they can be written with α + β and αβ and found without solving.

13m

Symmetric functions of the roots of a cubic or quartic

The same idea with three or four roots: Σα² = (Σα)² − 2Σαβ and Σ1/α = Σαβ/αβγ.

13m

Sums of powers of the roots

Every root satisfies the equation, so adding it over all the roots links each power sum to the ones before it.

13m

Problems with unknown coefficients

When the roots are linked, for example one is three times another, the relations give the missing coefficient.

13m

10.

Using a substitution: roots changed by a linear function

For roots α + 3 or 2α + 1, let y be the new root, write x in terms of y and put it into the old equation.

13m

11.

Using a substitution: reciprocal roots

Put x = 1/y and multiply through: the coefficients come out in reverse order, and the new equation answers questions about 1/α.

12m

12.

Using a substitution: squared roots

Put x = √y, collect the odd powers on one side and square both sides, so no square root is left.

14m

13.

Roots of polynomial equations: practice

A timed speed round, then eight mixed questions from the whole chapter, from reading off Σα to a squared-roots equation.

15m

14.

Roots of polynomial equations: exam practice

Three Paper 1 style questions with marks, full worked solutions and the mark scheme.

15m

15.

Roots of polynomial equations: summary

Every relation, identity and substitution from the chapter, 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.