On this page, we learn all about roots of polynomial equations and how to use the formulae for the sum of the roots and the product of the roots. For the polynomial equation, degree n
\(\sum _{ r=1 }^{ n }{ { a }_{ r } } { x }^{ r }\) , the sum of the roots = \(\frac { { a }_{ n-1 } }{ { a }_{ n } } \) , the product of the roots =\((-1)^n\frac { { a }_{ 0 } }{ { a }_{ n } } \)
Key Concepts
On this page, you should learn about
roots of polynomial equations
sums and products of roots
Essentials
The following videos will help you understand all the concepts from this page
Where do the formulae come from
Below you will find the formula for finding the sum of the roots and the product of the roots of polynomial equations of different degrees, but what are they and where do these formulae come from? We will find out in the video below.
* These formulae do not appear in the formula booklet. You should learn the general case for degree n by heart.
In the following video, we look at a typical exam-style question in which we are required to find a new polynomial having worked out the sum and product of roots of this equation:
The quadratic equation x² - 4x + 5 = 0 has roots \(\alpha\) and \(\beta\).
a. Without solving the equation, find the value of
i \(\alpha + \beta\)
ii \(\alpha \beta\)
b. Another quadratic equation 5x² + bx + c = 0 , b,c\(\in \mathbb{Z}\) has roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\).
The following video is a typical exam-style question. One of the things that IB likes to do in examinations, is to ask a question that requires knowledge from different areas of the course. This can be quite challenging and often something that textbooks do not give you practice in. This question is about sums and products of roots of polynomial equations, but requires some interesting problem-solving skils, since the roots of the equation are terms of an arithmetic sequence. Here is the question:
Consider the equation \(64x^{ 3 }−144x^{ 2 }+92x−15=0\)
Write down the numerical value of the sum and the product of the roots of this equation.
The roots of this equation are three consecutive terms of an arithmetic sequence. Solve the equation.
In the following video we will look at an example involving complex roots of a polynomial equation. Since the coefficients of the polynomial are real numbers, complex roots must always come in pairs and more than that they must be conjugate pairs - this way two complex numbers can multiply to give a real number.
The conjugate root theorem states that if the complex number \(a+ib\) is a root of a polynomial f(x) in one variable with real coefficients, then the complex conjugate\(a-ib\) also a root of that polynomial.
One root of the equation \(4z^4-4z^3-25z^2+55z-42=0\) is \(1+\frac {\sqrt{3}}{2}i\)
If \( \alpha=-\frac{5}{2} \quad \alpha+4=\frac{3}{2}\)
a(2x+5)(2x-3)=0 , a must be 1
4x²+4x-15=0
If \( \alpha=-\frac{3}{2} \quad \alpha+4=\frac{5}{2}\)
a(2x+3)(2x-5)=0 , a must be 1
4x²-4x-15=0
p>0
p=4
Exam-style Questions
Question 1
The quadratic equation \(3x^{ 2 }−8x+2=0\) has roots \(\alpha\) and \(\beta\).
a. Without solving the equation, find the value of \(\alpha + \beta\) and \(\alpha \beta\).
b. Another quadratic equation \(3x^{ 2 }+bx+c=0\quad ,\quad b,c\in \mathbb{Z}\) has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). Find the value of b and c.
Hint
b. Write the sum and product of \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\) in terms of \(\alpha + \beta\) and \(\alpha \beta\).
Consider the equation \(8x^{ 3 }−42x^{ 2 }+px−27=0.\)
State
the sum of the roots of the equation
the product of the roots of the equation
The roots of this equation are three consecutive terms of a geometric sequence. Taking the roots to be \(\frac{\alpha}{\beta},\alpha,\alpha\beta\) show that one of the roots is \(\frac{3}{2}\)
Solve the equation.
Find the value of p.
Hint
b. What is the product of the three roots \(\frac{\alpha}{\beta},\alpha,\alpha\beta\)?
c. What is the sum of the three roots \(\frac{\alpha}{\beta},\alpha,\alpha\beta\)?
d. The equation is \(a(x-\frac{\alpha}{\beta})(x-\alpha)(x-\alpha\beta)=0\)
The equation \(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\) has two real roots \(\alpha\) and \(\beta\) and two complex roots \(\gamma\) and \(\delta\) where \(\gamma=2-5i\).
a. Show that \(\alpha+\beta=\frac{1}{2}.\)
b. Find \(\alpha\beta\).
c. Hence find the two real roots α and β.
d. Find the values of p and q.
Hint
a. If γ=2−5i is a root…then δ = 2+5i is also a root. Work out γ+δ and sum of 4 roots
b. Work out γδ and the product of the 4 roots
c. Work out α and β using the two equations for \(\alpha+\beta\) and \(\alpha\beta\).
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