What happens if a term is missing, like in $x^3 + 4x - 2 = 0$?

A missing term simply means its coefficient is zero. In this case, the equation is $1x^3 + 0x^2 + 4x - 2 = 0$, so $a=1, b=0, c=4, d=-2$. This would mean, for example, that the sum of the roots $\Sigma \alpha = -b/a = -0/1 = 0$.

Do Vieta's formulas work if the roots are complex numbers?

Yes, absolutely. The relationships hold for all roots, whether they are real, rational, irrational or complex. This is a very powerful feature and provides a link to the topic of complex numbers.

How can I remember the signs for Vieta's formulas?

The signs simply alternate, starting with a negative. The sum of roots ($\Sigma \alpha$) is $-b/a$. The sum of products in pairs ($\Sigma \alpha\beta$) is $+c/a$. The sum of products in threes ($\Sigma \alpha\beta\gamma$) is $-d/a$, and so on. It's a simple, predictable pattern.

Is there a formula for $\Sigma \alpha^3$?

Yes, but it's more complex and less commonly required in exams than $\Sigma \alpha^2$. It can be derived from the identity $(\Sigma \alpha)(\Sigma \alpha^2) = \Sigma \alpha^3 + \dots$. A more direct method is to use the fact that each root satisfies the original equation. For a root $\alpha$, we have $a\alpha^3+b\alpha^2+c\alpha+d=0$. You can write this for $\beta$ and $\gamma$ too, and then sum the three equations: $a\Sigma\alpha^3 + b\Sigma\alpha^2 + c\Sigma\alpha + 3d = 0$. You can then rearrange for $\Sigma\alpha^3$ since you know all the other sums.

Where can I learn Roots of polynomial equations for Cambridge 9231?

This free lesson on MarkScheme covers syllabus point 1.1 (Roots of polynomial equations) for Cambridge A-Level Further Mathematics. It includes visual revision, key points, and links to real past-paper questions with mark-scheme marking at markscheme.app/courses/9231/1-1-roots-of-polynomial-equations.

Is Roots of polynomial equations (1.1) on 9231 free to study?

Yes. The Further Mathematics 9231 course on MarkScheme is 100% free — including premium visual lessons, exam tips, and past-paper marking. No paywall for Roots of polynomial equations.

Roots of polynomial equations 9231 — Free…

Roots of polynomial equations

Welcome to the study of the roots of polynomials. While you're familiar with solving quadratics, finding the roots of cubics and quartics can be incredibly complex. This topic provides a beautiful alternative: instead of finding the roots themselves, we can uncover deep relationships between them and the coefficients of the polynomial they belong to. These relationships, known as Vieta's formulas, are a cornerstone of classical algebra and a powerful tool in your Further Maths arsenal.

BY THE END, YOU CAN…

1Recall and use the relations between the roots and coefficients of polynomial equations up to degree 4.
2Evaluate symmetric sums of the roots, such as $\Sigma \alpha^2$ and $\Sigma \frac{1}{\alpha}$ .
3Form a new polynomial equation whose roots are a linear transformation of the roots of the original equation (e.g., $k\alpha, \alpha+k$ ).

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Secret Code of Polynomials

The coefficients of a polynomial equation hold a secret code about its roots. Vieta's formulas allow us to crack this code, revealing properties like the sum and product of roots without ever having to find them individually.

THINK OF IT LIKE…

Imagine a locked safe (the polynomial) and you don't have the keys (the roots). However, you're given clues on the outside of the safe (the coefficients) that tell you the total weight of all the gold bars inside (the sum of roots) and their combined volume (the product of roots). You can learn a lot about the contents without ever opening the safe.

1
Identify the polynomial's coefficients, ensuring it's in the form $ax^n + bx^{n-1} + \dots = 0$ .
2
Apply Vieta's formulas to find the fundamental sums of roots ( $\Sigma \alpha$ , $\Sigma \alpha\beta$ , etc.), watching the signs.
3
Identify the target expression you need to find (e.g., $\Sigma \alpha^2$ ) or the transformation for a new equation.
4
Use algebraic identities to express your target in terms of the fundamental sums, or perform a substitution to find a new equation.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$(\Sigma \alpha)^2$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Vieta's Formulas: The Core Relationships

For any polynomial equation, the coefficients are intrinsically linked to the sums and products of its roots. Vieta's formulas provide the exact expressions for these links. The key is to look at the coefficients in order, divide by the leading coefficient, and apply an alternating sign pattern.

Quadratic: For $ax^2+bx+c=0$ with roots $\alpha, \beta$ : $\Sigma \alpha = \alpha+\beta = -b/a$ $\alpha\beta = c/a$
Cubic: For $ax^3+bx^2+cx+d=0$ with roots $\alpha, \beta, \gamma$ : $\Sigma \alpha = \alpha+\beta+\gamma = -b/a$ $\Sigma \alpha\beta = \alpha\beta+\beta\gamma+\gamma\alpha = c/a$ $\alpha\beta\gamma = -d/a$
Quartic: For $ax^4+bx^3+cx^2+dx+e=0$ with roots $\alpha, \beta, \gamma, \delta$ : $\Sigma \alpha = -b/a$ $\Sigma \alpha\beta = c/a$ $\Sigma \alpha\beta\gamma = -d/a$ $\alpha\beta\gamma\delta = e/a$

EXAM TIP

Always check for missing terms in the polynomial. For example, in $2x^4 + 3x^2 - 5 = 0$ , the coefficients of $x^3$ and $x$ are both zero. So, $a=2, b=0, c=3, d=0, e=-5$ . This is a very common place to lose marks.

Evaluating New Symmetric Sums

The real power of Vieta's formulas is in finding the values of other symmetric expressions—expressions which remain unchanged if you swap the roots around. A classic example is finding the sum of the squares of the roots, $\Sigma \alpha^2$ . We can't get this directly from Vieta's formulas, but we can derive it using them.

Consider the expansion of $(\Sigma \alpha)^2$ . For a cubic with roots $\alpha, \beta, \gamma$ : $(\alpha+\beta+\gamma)^2 = \alpha^2+\beta^2+\gamma^2 + 2(\alpha\beta+\beta\gamma+\gamma\alpha)$ Rearranging gives the crucial identity: $\Sigma \alpha^2 = (\Sigma \alpha)^2 - 2(\Sigma \alpha\beta)$

Transformation of Roots

Another key skill is to find a new polynomial equation whose roots are related to the roots of an original equation. For example, if the original equation has roots $\alpha, \beta, \gamma$ , we might want to find the equation with roots $2\alpha, 2\beta, 2\gamma$ or $\alpha+3, \beta+3, \gamma+3$ . The method involves a clever substitution.

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

The equation $2x^3 - 7x^2 + 4x + 1 = 0$ has roots $\alpha, \beta, \gamma$ . Find the values of: (i) $\alpha + \beta + \gamma$ (ii) $\alpha\beta + \beta\gamma + \gamma\alpha$ (iii) $\alpha\beta\gamma$

1
First, identify the coefficients of the cubic $ax^3+bx^2+cx+d=0$ . Here, $a=2, b=-7, c=4, d=1$ .

EXAMPLE 2

The roots of the equation $x^4 + 2x^3 - 5x^2 + x - 3 = 0$ are $\alpha, \beta, \gamma, \delta$ . Find the value of $\Sigma \alpha^2$ .

1
The equation is a quartic with $a=1, b=2, c=-5, d=1, e=-3$ .

EXAMPLE 3

The equation $x^3 - 2x^2 + 5x - 4 = 0$ has roots $\alpha, \beta, \gamma$ . Find a cubic equation with integer coefficients whose roots are $\alpha+1, \beta+1, \gamma+1$ .

1
Let the original equation be $P(x) = x^3 - 2x^2 + 5x - 4 = 0$ , where $x$ can be $\alpha, \beta,$ or $\gamma$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEARoots of polynomial equations

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

For a general polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 = 0$ , what is the formula for the sum of the roots, $\Sigma \alpha$ ?

TAP OR SPACE TO FLIP

ANSWER

$\Sigma \alpha = -\frac{a_{n-1}}{a_n}$

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Quadratic: For $ax^2+bx+c=0$ with roots $\alpha, \beta$ : $\Sigma \alpha = \alpha+\beta = -b/a$ $\alpha\beta = c/a$
✓
Cubic: For $ax^3+bx^2+cx+d=0$ with roots $\alpha, \beta, \gamma$ : $\Sigma \alpha = \alpha+\beta+\gamma = -b/a$ $\Sigma \alpha\beta = \alpha\beta+\beta\gamma+\gamma\alpha = c/a$ $\alpha\beta\gamma = -d/a$
✓
Quartic: For $ax^4+bx^3+cx^2+dx+e=0$ with roots $\alpha, \beta, \gamma, \delta$ : $\Sigma \alpha = -b/a$ $\Sigma \alpha\beta = c/a$ $\Sigma \alpha\beta\gamma = -d/a$ $\alpha\beta\gamma\delta = e/a$

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

NEXT →1.2 Rational functions and graphs

Roots of polynomial equations

BY THE END, YOU CAN…

1Recall and use the relations between the roots and coefficients of polynomial equations up to degree 4.
2Evaluate symmetric sums of the roots, such as $\Sigma \alpha^2$ and $\Sigma \frac{1}{\alpha}$ .
3Form a new polynomial equation whose roots are a linear transformation of the roots of the original equation (e.g., $k\alpha, \alpha+k$ ).

In simple terms

A friendly intro before the formal notes — no formulas yet.

The Secret Code of Polynomials

THINK OF IT LIKE…

1
Identify the polynomial's coefficients, ensuring it's in the form $ax^n + bx^{n-1} + \dots = 0$ .
2
Apply Vieta's formulas to find the fundamental sums of roots ( $\Sigma \alpha$ , $\Sigma \alpha\beta$ , etc.), watching the signs.
3
Identify the target expression you need to find (e.g., $\Sigma \alpha^2$ ) or the transformation for a new equation.
4
Use algebraic identities to express your target in terms of the fundamental sums, or perform a substitution to find a new equation.

Explore the concept

Use the live diagram and synced steps — play it or tap a step card to walk through.

Key formulas

Tap any symbol to reveal exactly what it means and its units.

$(\Sigma \alpha)^2$

Tap a symbol — great for exam definitions

Full topic notes

Formal explanation with the rigour you need for the exam.

Vieta's Formulas: The Core Relationships

Quadratic: For $ax^2+bx+c=0$ with roots $\alpha, \beta$ : $\Sigma \alpha = \alpha+\beta = -b/a$ $\alpha\beta = c/a$
Cubic: For $ax^3+bx^2+cx+d=0$ with roots $\alpha, \beta, \gamma$ : $\Sigma \alpha = \alpha+\beta+\gamma = -b/a$ $\Sigma \alpha\beta = \alpha\beta+\beta\gamma+\gamma\alpha = c/a$ $\alpha\beta\gamma = -d/a$
Quartic: For $ax^4+bx^3+cx^2+dx+e=0$ with roots $\alpha, \beta, \gamma, \delta$ : $\Sigma \alpha = -b/a$ $\Sigma \alpha\beta = c/a$ $\Sigma \alpha\beta\gamma = -d/a$ $\alpha\beta\gamma\delta = e/a$

EXAM TIP

Evaluating New Symmetric Sums

Transformation of Roots

Worked examples

See the formulas applied — reveal one step at a time, like the exam.

EXAMPLE 1

The equation $2x^3 - 7x^2 + 4x + 1 = 0$ has roots $\alpha, \beta, \gamma$ . Find the values of: (i) $\alpha + \beta + \gamma$ (ii) $\alpha\beta + \beta\gamma + \gamma\alpha$ (iii) $\alpha\beta\gamma$

1
First, identify the coefficients of the cubic $ax^3+bx^2+cx+d=0$ . Here, $a=2, b=-7, c=4, d=1$ .

EXAMPLE 2

The roots of the equation $x^4 + 2x^3 - 5x^2 + x - 3 = 0$ are $\alpha, \beta, \gamma, \delta$ . Find the value of $\Sigma \alpha^2$ .

1
The equation is a quartic with $a=1, b=2, c=-5, d=1, e=-3$ .

EXAMPLE 3

The equation $x^3 - 2x^2 + 5x - 4 = 0$ has roots $\alpha, \beta, \gamma$ . Find a cubic equation with integer coefficients whose roots are $\alpha+1, \beta+1, \gamma+1$ .

1
Let the original equation be $P(x) = x^3 - 2x^2 + 5x - 4 = 0$ , where $x$ can be $\alpha, \beta,$ or $\gamma$ .

How it all connects

The big idea sits in the middle — tap a linked idea to explore the link.

MAIN IDEARoots of polynomial equations

Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.

Glossary

Try to recall each definition before you reveal it.

Quick check

Answer in your head first — then tap to check. No pressure.

Revision flashcards

Flip the card. Test yourself before the exam.

QUESTION · 1 / 12

For a general polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 = 0$ , what is the formula for the sum of the roots, $\Sigma \alpha$ ?

TAP OR SPACE TO FLIP

ANSWER

$\Sigma \alpha = -\frac{a_{n-1}}{a_n}$

TAP OR SPACE TO FLIP BACK

1 / 12 · ← → space

Key takeaways

Review these before you close the topic — retrieval beats re-reading.

✓
Quadratic: For $ax^2+bx+c=0$ with roots $\alpha, \beta$ : $\Sigma \alpha = \alpha+\beta = -b/a$ $\alpha\beta = c/a$
✓
Cubic: For $ax^3+bx^2+cx+d=0$ with roots $\alpha, \beta, \gamma$ : $\Sigma \alpha = \alpha+\beta+\gamma = -b/a$ $\Sigma \alpha\beta = \alpha\beta+\beta\gamma+\gamma\alpha = c/a$ $\alpha\beta\gamma = -d/a$
✓
Quartic: For $ax^4+bx^3+cx^2+dx+e=0$ with roots $\alpha, \beta, \gamma, \delta$ : $\Sigma \alpha = -b/a$ $\Sigma \alpha\beta = c/a$ $\Sigma \alpha\beta\gamma = -d/a$ $\alpha\beta\gamma\delta = e/a$

Practice — then mark it

The whole point: a real Cambridge question, marked mark-by-mark.

Real past paper[0]

Practice Questions

Do it on paper → mark itMARKED MARK-BY-MARK · B1 / M1 / A1 · OFFICIAL SCHEME

Extra simulations & links

PhET, GeoGebra and other curated tools — open in a new tab.

Frequently asked

✓

Checkpoint

One marked question is worth ten re-reads — close the loop before you move on.

Checkpoint

Reading it isn’t knowing it — prove it.

Before you move on: do Practice Questions on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.

Attempt & get marked →Takes about a minute · you’ll land back on this lesson

NEXT →1.2 Rational functions and graphs

Revise Roots of polynomial equations for Cambridge 9231 (1.1)

In simple terms

Explore the concept

Key formulas

Full topic notes

Vieta's Formulas: The Core Relationships

Evaluating New Symmetric Sums

Transformation of Roots

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

In simple terms

Explore the concept

Key formulas

Full topic notes

Vieta's Formulas: The Core Relationships

Evaluating New Symmetric Sums

Transformation of Roots

Worked examples

How it all connects

Glossary

Quick check

Revision flashcards

Key takeaways

Practice — then mark it

Practice Questions

Extra simulations & links

Frequently asked

Checkpoint

Reading it isn’t knowing it — prove it.

Discuss Roots of polynomial equations