Mathematics · Class 12 Science
Ch 9Theory of Equations — Class 12 Mathematics, concept-first.
An algebraic equation in one variable is an equation of the form , where is a polynomial of degree (so ). The degree of the equation is , the highest power of occurring in it. When the equation is called a monic equation.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Forming an Equation from Given Roots
To build the monic polynomial equation whose roots are a prescribed list of numbers , form the product and expand it.
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Relation between Roots and Coefficients
An algebraic equation in one variable is an equation of the form , where is a polynomial of degree (so ). The degree of the equation is , the highest power of occurring in it.
Polynomial Equations and the Remainder Theorem
Before relating roots to coefficients we need one basic tool: the Remainder Theorem. It says that if a polynomial is divided by the linear polynomial , the remainder is always a constant, and that con…
Relations between Roots and Coefficients
Suppose are the three roots of the cubic . Since a monic polynomial with these roots must equal , we can expand this product and compare it, coefficient by coefficient, with : Matching coefficients gi…
+−Exercise 4(a)i8 questions
- Q1Form the monic polynomial equation of degree $3$ whose roots are $2, 3, 6$.Free
- Q2If $\alpha,\beta,\gamma$ are the roots of $x^3+2x^2-3x-1=0$, find the values of $\alpha+\beta+\gamma$, $\alpha\beta+\beta\gamma+\gamma\alpha…Free
- Q3If $\alpha,\beta,\gamma$ are the roots of $x^3+px^2+qx+r=0$, express $\dfrac{1}{\alpha^2}+\dfrac{1}{\beta^2}+\dfrac{1}{\gamma^2}$ in terms o…Free
- Q4Solve $x^3-7x^2+14x-8=0$, given that its roots are in geometric progression.Preview
- Q5Solve $8x^3-36x^2+46x-15=0$, given that its roots are in arithmetic progression.Preview
- Q6Find $\alpha^2+\beta^2+\gamma^2$, where $\alpha,\beta,\gamma$ are the roots of $x^3-6x^2+11x-6=0$, without solving the equation.Preview
- Q7If the roots of $x^3+ax^2+bx+c=0$ are in harmonic progression, prove that $2b^3-9abc+27c^2=0$.Preview
- Q8Form the monic cubic equation with rational coefficients, two of whose roots are $2+\sqrt3$ and $2-\sqrt3$, the third root being $1$.Preview
Forming an Equation from Given Roots
The relations of the previous section run just as well in reverse: given a prescribed list of numbers that we want to be the roots of an equation, the monic equation having exactly these roots is Expa…
Roots in Arithmetic, Geometric and Harmonic Progression
Sometimes a problem does not simply hand us the roots of a cubic -- it tells us the roots satisfy an extra structural condition, such as being in arithmetic, geometric, or harmonic progression, and as…
Solving Equations Whose Roots Satisfy an Extra Relation
Locating the roots of a specific numerical equation, or exploiting a known relation among them, calls for two complementary algebraic tools: synthetic division, which both tests candidate roots and re…
Synthetic Division and the Division Algorithm
The division algorithm for polynomials states that, given and a nonzero divisor , there exist unique polynomials (quotient) and (remainder), with , such that .
+−Exercise 4(b)i4 questions
- Q1Using synthetic division, find the quotient and remainder when $f(x)=2x^4-3x^3+4x^2-5x+6$ is divided by $x-1$.Free
- Q2Find the multiple roots of $x^4-6x^3+13x^2-12x+4=0$ by the H.C.F. method.Free
- Q3Divide $f(x)=x^4-3x^3+2x^2+5x-7$ by $x-2$ using synthetic division, and state the quotient and the remainder.Preview
- Q4Find all the roots of $x^3-6x^2+11x-6=0$ by first locating one root by trial and error.Preview
Multiple Roots and the H.C.F. Method
A root of is called a multiple root of order (or a root of multiplicity ) if with . Differentiating this factorisation shows , and since the bracket does not vanish at (it equals ), the root survives…
Nature of the Roots of an Equation with Real Coefficients
So far the roots we have manipulated could, in principle, be any complex numbers. But when an equation's coefficients are constrained -- to be real, or more strongly to be rational -- its non-real and…
Complex Conjugate Roots
Theorem (Complex Conjugate Roots). If is an equation with real coefficients and (with ) is a root, then its conjugate is also a root, with the same multiplicity.
Irrational Conjugate Roots
Theorem (Irrational Conjugate Roots). If is an equation with rational coefficients and (with rational and irrational) is a root, then its conjugate is also a root, with the same multiplicity.
+−Exercise 4(c)i4 questions
- Q1Given that one root of $x^3-6x^2+13x-10=0$ is $2$, find all the roots of the equation.Free
- Q2Given that $1+2i$ is a root of $x^4-4x^3+6x^2-4x-15=0$, find all the roots.Free
- Q3Given that $2+\sqrt5$ is a root of $x^3-2x^2-9x-2=0$, find all the roots.Preview
- Q4Form the monic cubic equation with rational coefficients having $3-\sqrt2$ and $4$ among its roots.Preview
Transformation of Equations
It is often useful to convert an equation into a new equation whose roots are related to the original roots by some fixed rule -- negated, scaled by a constant, shifted, inverted, or squared -- withou…
Basic Transformations: Sign, Scale and Translation
Four elementary substitutions cover most of what is needed:
Reciprocal Equations
The fifth standard transformation replaces each root by its reciprocal: if is a root of the degree- equation , then is a root of , which amounts simply to writing the original coefficients in reverse…
+−Exercise 4(d)i8 questions
- Q1Find the equation whose roots are the negatives of the roots of $x^4+3x^3-6x^2-5x+3=0$.Free
- Q2Find the equation whose roots are twice the roots of $x^3-6x^2+11x-6=0$.Free
- Q3Using synthetic division, find the equation whose roots are $2$ less than the roots of $x^3-6x^2+10x-3=0$.Free
- Q4Find the equation whose roots are the reciprocals of the roots of $x^3-6x^2+11x-6=0$.Preview
- Q5Find the equation whose roots are the squares of the roots of $x^3-6x^2+11x-6=0$.Preview
- Q6Solve the reciprocal equation $x^4-2x^3-x^2-2x+1=0$.Preview
- Q7Solve the reciprocal equation $x^5-5x^4+9x^3-9x^2+5x-1=0$.Preview
- Q8Remove the second term from the equation $x^3+6x^2+3x-1=0$.Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 12 questionsHide questions12 questions
- Q1If the product of roots of $4x^3 + 16x^2 - 9x - a = 0$ is 9, then find $a$.Preview
- Q2Find the algebraic equation whose roots are the translates of the roots of the equation $x^4 - 5x^3 + 7x^2 - 17x + 11 = 0$ by $-2$.Preview
- Q3If $1, 1, \alpha$ are the roots of $x^3-6x^2+9x-4=0$, then find $\alpha$.Preview
- Q4Solve $18x^3+81x^2+121x+60=0$ given that one root is equal to half the sum of the remaining roots.Preview
- Q5If $1, 1, \alpha$ are the roots of $x^3 - 6x^2 + 9x - 4 = 0$ then find $\alpha$.Preview
- Q6Solve the following equation $x^4 - 10x^3 + 26x^2 - 10x + 1 = 0$.Preview
- Q7If $1, -2$ and $3$ are the roots of $x^3 - 2x^2 + ax + 6 = 0$, then find $a$.Preview
- Q8Solve the equation $3x^3 - 26x^2 + 52x - 24 = 0$ given that the roots are in G.P.Preview
- Q9If $\alpha, \beta, \gamma$ are the roots of $4x^3 - 6x^2 + 7x + 3 = 0$, then find the value of $\alpha\beta + \beta\gamma + \gamma\alpha$.Preview
- Q10Find the polynomial equation whose roots are the translates of those of the equation $x^5 - 4x^4 + 3x^2 - 4x + 6 = 0$ by $-3$.Preview
- Q11If $1, 1, \alpha$ are the roots of $x^3 - 6x^2 + 9x - 4 = 0$, then find $\alpha$.Preview
- Q12Solve the equation $8x^3 - 36x^2 - 18x + 81 = 0$, given that the roots are in Arithmetic Progression.Preview