Polynomials Class 9 Maths Chapter 2 Notes

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Last Updated on December 13, 2024 by XAM CONTENT

Master the fundamentals of the Polynomials with our Class 9 Maths Chapter 2 notes, covering essential concepts, solved examples, and key points to help you excel in your exams

Chapter	Polynomials
Type of Questions	Notes
Nature of Articles	Notes with examples
Board	CBSE
Class	9
Subject	Maths
Useful for	Class 9 Studying Students
Important Link	Class 9 Maths Chapterwise Notes

Polynomials Notes

➤ Algebraic Expression: Any expression that contains constants and variables, connected by some or all of the operations,,$+- \times$ and $\therefore$ is known as an algebraic expression. For example, $x^2+1,6 x^2-5 y^2+2 x y$ etc.

➤ Polynomials: An algebraic expression in which the variables involved have only non-negative integral powers. For example, $7 x+y+5, a+b$, etc.

➤ Polynomial in One Variable: An algebraic expression which consists of only one type of variable in the entire expression. For example, $2 x^2+5 x-7,3 y^3+12 y^2+7 y-9$, etc.

➤ General Expression of Polynomial: A polynomial in one variable $x$ of degree $n$ can be expressed as

$$
a_n x^n+a_{n-1} x^{n-1}+\ldots+a_1+a_0
$$

where, $a_n \neq 0, a_0$ is constant term and $a_1, a_2, \ldots . . ., a_n$ are called coefficients of $x, x^2, x^3, \ldots x^n$ respectively.

➤ Terms and their Coefficients:
If $f(x)=a_n x^n+$ $a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+\ldots .+a_1 x+a_0$ is a polynomial in variable $x$, then $a_n x^n, a_{n-1} x^{n-1}, a_{n-2} x^{n-2}, \ldots ., a_1 x$ and $a_0$ are known as the terms of polynomial $f(x)$ and $a_n, a_{n-1}, a_{n-2}, \ldots, a_1$ and $a_0$ respectively are known as their coefficients.

➤ Degree of a Polynomial: Highest power of a variable in the polynomial is a degree of the polynomial. For example, in case of $6 x^4+5 x^3+3$, the highest power of $x$ is 4 , so the degree of polynomial is 4 .

➤ Types of Polynomials:
1. Linear Polynomial: Polynomial of degree 1. In general, $a x+b, a \neq 0$ is a linear polynomial.
2. Quadratic Polynomial: Polynomial of degree 2. In general, $a x^2+b x+c, a \neq 0$ is a quadratic polynomial.
3. Cubic Polynomial: Polynomial of degree 3. In general, $a x^3+b x^2+c x+d, a \neq 0$ is a cubic polynomial.
4. Biquadratic Polynomial: Polynomial of degree 4. In general, $a x^4+b x^3+c x^2+d x+e, a \neq 0$ is a biquadratic polynomial.
5. Constant Polynomial: Polynomial of degree 0 , consisting of a non-zero constant.
For example, 12, $-7,9 / 14$, etc.
6. Zero Polynomial: A polynomial consisting of one term, namely zero.

➤ Classification of Polynomials:
1. Monomial: Polynomial with only one term. For example, $5 x, \frac{3}{8} y$, etc.
2. Binomial: Polynomial having two terms. For example, $8 x+5 x,-7 y^2+8 y$ etc.
3. Trinomial: Polynomial having three terms.

For example, $3 y^2+4 y+\frac{19}{7}$ etc.

Things to Remember:

1. Linear polynomial can be monomial or binomial.
2. Quadratic polynomial can be monomial, binomial or trinomial.
3. The degree of a zero polynomial is not defined.

Zeroes of a Polynomial: Let $p(x)$ be a polynomial in one variable and ‘ $\alpha$ ‘ be a real number such that the value of polynomial at $x=\alpha$ is zero, i.e., $p(\alpha)=0$, then ‘ $\alpha$ ‘ is said to be a zero of a polynomial $p(x)$.

Things to Remember:

1. A non-zero constant polynomial has no zero.
2. 0 may or may not be the zero(s) of a given polynomial.
3. A polynomial of nth degree can have maximum n zeroes.

➤ Division Algorithm in Polynomials: Suppose $p(x)$ and $g(x)$ are two polynomials such that degree $p(x) \geq$ degree $g(x)$. When we divide $p(x)$ by $g(x)$, then we get the result in the form of

$$
p(x)=g(x) \cdot q(x)+r(x)
$$

where $q(x)=$ quotient
and $\quad r(x)=$ remainder

➤ Remainder Theorem: Let $p(x)$ be a polynomial having degree 1 or more than 1 and let $\alpha$ be any real number. If $p(x)$ is divided by $(x-\alpha)$, then remainder is $p(\alpha)$.

➤ Factor Theorem: Suppose $p(x)$ be a polynomial of degree 1 or more than 1 and $\alpha$ be any real number.
(i) If $p(\alpha)=0$, then $(x-\alpha)$ is a factor of $p(x)$.
(ii) If $(x-\alpha)$ is a factor of $p(x)$, then $p(\alpha)=0$.

➤ Factorisation of Quadratic Polynomial: Quadratic polynomial can be factorised either by splitting middle term or by using factor theorem.

1. Polynomial of the form $\boldsymbol{x}^2+\boldsymbol{b} \boldsymbol{x}+\boldsymbol{c}$ : We find integers $p$ and $q$ such that $p+q=b$ and $p q=c$.
Then, $x^2+b x+c=x^2+(p+q) x+p q$

$$
\begin{aligned}
& =x^2+p x+q x+p q \\
& =x(x+p)+q(x+p) \\
& =(x+p)(x+q)
\end{aligned}
$$

2. Polynomial of the form $\boldsymbol{a} \boldsymbol{x}^2+\boldsymbol{b x}+\boldsymbol{c}$ : We find integers $p$ and $q$ such that $p+q=b$ and $p q=a c$.
Then, $a x^2+b x+c=a x^2+(p+q) x+\frac{p q}{a}$

$$
\begin{aligned}
& =\frac{a^2 x^2+a p x+a q x+p q}{a} \\
& =\frac{a x(a x+p)+q(a x+p)}{a} \\
& =-(a x+p)(a x+q)
\end{aligned}
$$

➤ Factorisation of Cubic Polynomial: To factorise a cubic polynomial $\rho(x)$, we
(i) find $x=a$, where $p(a)=0$
(ii) then $(x-a)$ is a factor of $p(x)$.
(iii) now, divide $p(x)$ by $(x-a)$ i.e., $p(x) /(x-a)$.
(iv) and then we factorise the quotient polynomial by splitting the middle term.

➤ Algebraic Identities: Algebraic equations that are true for all values of variables occurring in it. Some useful algebraic identities are:
(i) $(x+y)^2=x^2+2 x y+y^2$
(ii) $(x-y)^2=x^2-2 x y+y^2$
(iii) $x^2-y^2=(x+y)(x-y)$
(iv) $(x+a)(x+b)=x^2+(a+b) x+a b$
(v) $(x+y+z)^2$
$=x^2+y^2+z^2+2 x y+2 y z+2 z x$
(vi) $(x+y)^3=x^3+y^3+3 x y(x+y)$
$=x^3+y^3+3 x^2 y+3 x y^2$
(vii) $(x-y)^3=x^3-y^3-3 x y(x-y)$
$=x^3-y^3-3 x^2 y+3 x y^2$
(viii) $x^3+y^3=(x+y)\left(x^2-x y+y^2\right)$
$=(x+y)^3-3 x y(x+y)$
(ix) $x^3-y^3=(x-y)\left(x^2+x y+y^2\right)$
$=(x-y)^3+3 x y(x-y)$
(x) $\left(x^3+y^3+z^3-3 x y z\right)$
$=(x+y+z)\left(x^2+y^2+z^2 -x y-y z-z x)\right.$
If $x+y+z=0$, then $x^3+y^3+z^3=3 x y z$.

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