Sum & Product of Roots of Polynomials (DP IB Analysis & Approaches (AA): HL): Revision Note

Sum & product of roots

How do I find the sum & product of the roots of a polynomial?

• Suppose  $P\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+ \dots +a_{1}x+a_0$ is a polynomial of degree n

  • $a_n$  is the coefficient of the leading term

  • $a_{n-1}$  is the coefficient of the $x^{n-1}$ term

    • This could be equal to zero

    • e.g. $P\left(x\right)=3x^4+x^2-x+1$

  • $a_0$ is the constant term

    • This could be equal to zero

    • e.g. $P\left(x\right)=2x^3-5x^2+3x$

• The n roots of the equation $P\left(x\right)=0$ are denoted as ${\alpha }_1,{\alpha }_2,...,{\alpha }_n$

  • Some roots might be complex and/or repeated

  • You can find their sum and product without finding the values of the roots

• In factorised form $P\left(x\right)=a_n\left(x-{\alpha }_1\right)\left(x-{\alpha }_2\right)...\left(x-{\alpha }_n\right)$

  • The coefficient of the $x^{n-1}$ term is $a_n\left(-{\alpha }_1-{\alpha }_2-...-{\alpha }_n\right)$

  • The constant term is $a_n\left(-{\alpha }_1\right)\times \left(-{\alpha }_2\right)\times ...\times \left(-{\alpha }_n\right)$

• The sum of the roots is given by:

  •  ${\alpha }_1+{\alpha }_2+\dots +{\alpha }_n=-\frac{a_{n-1}}{a_n}$

• The product of the roots is given by:

  • ${\alpha }_1\times { \alpha }_2\times \dots \times {\alpha }_n=\frac{{\left(-1\right)}^n{a}_0}{a_n}$

For example, consider $5x^4+2x^3-3x^2+x-7=0$

• The sum of the roots is equal to $-\frac{2}{5}$

• The product of the roots is equal to $\frac{{\left(-1\right)}^4\times \left(-7\right)}{5}=-\frac{7}{5}$

How can I find unknowns if I am given the sum and/or product of the roots of a polynomial?

• Write down all the roots you know

  • If you know a complex root of a real polynomial then its complex conjugate is another root

• You can form two equations using the roots

  • One using the sum of the roots formula

  • One using the product of the roots formula

• Solve the equations to find any unknowns