-
Engineering and Architecture
Exams
Colleges
Predictors
Resources
-
Computer Application and IT
Quick Links
Colleges
-
Pharmacy
Colleges
Resources
-
Hospitality and Tourism
Colleges
Resources
Diploma Colleges
-
Competition
Other Exams
Resources
-
School
Exams
Top Schools
Products & Resources
-
Study Abroad
Top Countries
Resources
-
Arts, Commerce & Sciences
Colleges
Upcoming Events
Resources
-
Management and Business Administration
Exams
Colleges & Courses
Predictors
-
Learn
Law Preparation
MBA Preparation
Engineering Preparation
Medical Preparation
-
Online Courses and Certifications
Top Streams
Specializations
- Digital Marketing Certification Courses
- Cyber Security Certification Courses
- Artificial Intelligence Certification Courses
- Business Analytics Certification Courses
- Data Science Certification Courses
- Cloud Computing Certification Courses
- Machine Learning Certification Courses
- View All Certification Courses
Resources
-
Medicine and Allied Sciences
Colleges
Predictors
Resources
-
Law
Resources
Colleges
-
Animation and Design
Exams
Predictors & Articles
Colleges
Resources
-
Media, Mass Communication and Journalism
Colleges
Resources
-
Finance & Accounts
Top Courses & Careers
Colleges
Polynomial Equation of Higher Degree, Remainder theorem - (Concept)
Equation of higher degree
is known as the polynomial equation of degree n which has exactly n roots (i.e., number of real roots + number of imaginary roots = n)
Relation between its coefficients and roots
$
\text { sum of all roots }=\sum \alpha_1=\alpha_1+\alpha_2+\ldots+\alpha_{n-1}+\alpha_n=(-1) \frac{\mathrm{a}_1}{\mathrm{a}_0}
$
sum of products taken two at a time
$
\sum \alpha_1 \alpha_2=\alpha_1 \alpha_2+\alpha_1 \alpha_3+\ldots+\alpha_1 \alpha_{\mathrm{n}}+\alpha_2 \alpha_3+\ldots+\alpha_2 \alpha_{\mathrm{n}}+\ldots+\alpha_{\mathrm{n}-1} \alpha_{\mathrm{n}}=(-1)^2 \frac{a_2}{a_0}
$
sum of products taken three at a time
$
\sum \alpha_1 \alpha_2 \alpha_3=(-1)^3 \frac{\mathrm{a}_3}{\mathrm{a}_0}
$
product of all roots $=\alpha_1 \alpha_2 \ldots \alpha_n=(-1)^n \frac{a_n}{a_0}$
For example,
Suppose $\mathrm{n}=3$ and $a x^3+b x^2+c x+d=0$ is polynomial equation with $\mathrm{a} \neq 0$ and ?,? and $?$ are the roots of the equation then :
$
\begin{aligned}
& \alpha+\beta+\gamma=-\frac{b}{a} \\
& \sum \alpha \beta=\alpha \beta+\beta \gamma+\gamma \alpha=(-1)^2 \frac{\mathrm{c}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{a}} \\
& \alpha \beta \gamma=(-1)^3 \frac{d}{\mathrm{a}}=-\frac{d}{\mathrm{a}}
\end{aligned}
$
Transformation of roots
For transformation of roots, we can use the same procedure we used in case of quadratic equations.
Remainder theorem
The remainder theorem states that if a polynomial f(x) is divided by a linear function (x - k), then the remainder is f(k).
In Division,
Dividend = Divisor x Quotient + Remainder
For polynomials also we can use this theorem
f(x) = d(x).q(x) + r(x)
where f(x) is the divisor, d(x) is the divisor, q(x) is the quotient and r(x) is the remainder. And these 4 are polynomials
Degree of remainder r(x) is always less than degree of divisor d(x)
Now, if divisor d(x) is a linear polynomial (x-k). Let q(x) be the quotient, remainder r(x) will be a constant value equal to R:
f(x) = (x - k)q(x) + R
Now if we put x = k
i.e. f(k) = (k - k)q(x) + R = 0 + R
f(k) = R
So, remainder is f(k), when f(x) is divided by a linear polynomial (x-k)
Eg, To find remainder when f(x) = 2x3 - 3x - 4 is divided by (x-3),
Here k = 3, So remainder will be f(k) = f(3) = 2.(3)3 - 3(3) - 4 = 54 - 9 - 4 = 41
Factor Theorem
Now if f(k) = 0, then this means that remainder when f(x) is divided by (x-k) is 0.
As remainder is 0, so (x-k) is a factor of f(x)
So, factor theorem states that if f(k) - 0, then (x-k) is a factor of f(x).
Eg, f(x) = x3 + 3x - 4
Now we can observe by hit and trial that f(1) = 1 + 3 - 4 = 0, so (x-1) is a factor of f(x).
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
If is one of the roots of the equation,
then the real root of this equation:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the real roots of the equation
$\begin{vmatrix} x & -6 &-1 \\ 2 &-3x &x-3 \\ -3& 2x &x+2 \end{vmatrix}=0$, is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let are two roots of
& satisfies
, if
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
has roots 2 and 3 then m+n equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
and
are roots of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be the roots of the equation
If
then which one of the following statements is not true?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be the real roots of the equation,
If the system of equations
given by
has non-trivial solution, then the value of
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let are three roots of
such that
then ( it is given
)
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by (x−1) and it leaves remainder 6 when divided by (x+1); then :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If,$\mathrm{a}+\mathrm{b}+\mathrm{c}=1$, $\mathrm{ab}+\mathrm{bc}+\mathrm{ca}=2$ and $\mathrm{abc}=3$ then the value of$a^4+b^4+c^4$ is equal to_________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of distinct real roots of the equation $3 x^{4}+4 x^{3}-12 x^{2}+4=0$ is ___________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The set of all values of for which the equation
has real roots is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The function $f(x)=x^3-6 x^2+a x+b$ is such that $f(2)=f(4)=0$. Consider two statements.
(S1) there exists $x_1, x_2 \in(2,4), x_1<x_2$, such that $f^{\prime}\left(x_1\right)=-1$ and $f^{\prime}\left(x_2\right)=0$. (S2) there exists $x_3, x_4 \in(2,4), x_3<x_4$, such that $f$ is decreasing in $\left(2, x_4\right)$, increasing in $\left(x_4, 4\right)$ and $2 f^{\prime}\left(x_3\right)=\sqrt{3} f\left(x_4\right)$.
Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be a polynomial of degree 3 such that
. Then the value of
is equal to________
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following are zeros of the equation
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of n such that
is a prime number is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following is not a polynomial:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Remainder when is divided by
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Remainder when is divided by
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following cannot be the remainder when is divided by
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If x = 1,2,3,4 satisfies the equation then the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are common roots between equations
and
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
have two common roots. Then the modulus of sum of their third roots is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
have two common roots then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of the equation
, then the value of
is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\forall n \in \mathbb{N}$, if $37^{(n+2)}+16^{(n+1)}+30^n$ is divided by 7 , then the remainder is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Sum of product of pain of root in cubic equation $kx^{3}+2x^{2}-kx+1=0$ is -1 for what value of k
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$
\text { What is the sum of the root of the equation } 2 x^3-x+1=0 ?
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is the value of 'a' so that the product of roots of the cubic equation is 1, where the cubic equation is:
$
a x^3+b x^2-2 x+7=0
$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta$ be roots of $x^2+\sqrt{2} x-8=0$. If $\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}$, then $\frac{\mathrm{U}_{10}+\sqrt{12} \mathrm{U}_9}{2 \mathrm{U}_8}$ is equal to___________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f(x)=x^{12}-x^9+x^4-x+1$. Which of the following is true?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
These are how many roots of quadratic equation : $k x^2-2 x+7=0$ between $(-1,2)$ such that $f(-1)$ $f(2)<0$, where $f(x)=k x^2-2 x+7$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following option is correct for the equation $2 x^3-15 x^2+37 x-30=0$, where $\alpha, \beta, \gamma$ are the roots of the equation and roots are in AP.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a=\cos 1^{\circ}$, and $b=\sin 1^{\circ}$. We say that a real number is algebraic if it is a root of a polynomial with integer coefficients. Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\mathrm{a}_0=0$ and $\mathrm{a}_{\mathrm{x}}=3 \mathrm{a}_{\mathrm{n}-1}+1$ for $\mathrm{n} \geq 1$. Then the remainder obtained dividing $\mathrm{a}_{2010}$ by 11 is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let p(x)=a0+a1x+......+anxn be a non-zero polynomial with integer coefficients. If $\small p\left ( \sqrt{2}+\sqrt{3}+\sqrt{6} \right )=0$, the smallest possible value of n is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f(x)=x^6-2 x^5+x^3+x^2-x-1 \ {\text {and }} g(x)=x^4-x^3-x^2-1$ be two polynomials. Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$, and d be the roots of $g(x)=0$. Then the value of $f(a)+f(b)+f(c)+f(d)$ is
(A) -5
(B) 0
(C) 4
(D) 5
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The quotient when $1+x^{2}+x^{4}+x^{6}+.......................+x^{34}$ is divided by $1+x+x^{2}+x^{3}+.......................+x^{17}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The remainder when the polynomial $1+x^{2}+x^{4}+x^{6}+.............+x^{22}$ is divided by$1+x+x^{2}+x^{3}+.............+x^{11}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of non-real roots of the polynomial equation $x^{3}+3x^{2}+3x+3=0$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of zeroes of the polynomial $ x^{4}-2x^{3}-7x^{2}+20x-12$ are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $n>2$ be an integer and define a polynomial $p(x)=x^n+a_{n-1} x^{n-1}+\ldots+a_1 x+a_0$
where $a_0, a_1, \ldots a_{n-1}$ are integers. Suppose we know that $n p(x)=(1=x) p^{\prime}(x)$.If $b=p(1)$, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
