Chapter 2: Polynomials

Class 9 Maths Chapter 2 on Polynomials includes definitions, the Remainder Theorem, Factor Theorem, and algebraic expressions. These questions are very important for examinations.

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1. If $x^{51}+51$ is divided by $x+1$, the remainder is

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2. Zero of the zero polynomial is

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3. If $p(x)=x+3$, then $p(x)+p(-x)$ is equal to

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4. If $p(x)=x^{2}-2 \sqrt{2} x+1$, then $p(2 \sqrt{2})$ is equal to

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5. The factorisation of $4 x^{2}+8 x+3$ is

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6. If $a+b+c=0$, then $a^{3}+b^{3}+c^{3}$ is equal to

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7. If $\frac{x}{y}+\frac{y}{x}=-1 \quad(x, y \neq 0)$, the value of $x^{3}-y^{3}$ is

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8. If $49 x^{2}-b=\left(7 x+\frac{1}{2}\right)\left(7 x-\frac{1}{2}\right)$, then the value of $b$ is

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9. Which of the following is a factor of $(x+y)^{3}-\left(x^{3}+y^{3}\right)$ ?

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10. One of the factors of $\left(25 x^{2}-1\right)+(1+5 x)^{2}$ is

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11. Which one of the following is a polynomial?

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12. Zero of the polynomial $p(x)=2 x+5$ is

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13. $\sqrt{2}$ is a polynomial of degree

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14. The coefficient of $x$ in the expansion of $(x+3)^{3}$ is

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15. If $x^{2}+k x+6=(x+2)(x+3)$ for all $x$, then the value of $k$ is

Important Definitions and Theorems

Polynomial:
A polynomial is a mathematical expression that consists of variables and coefficients. It can have one or more terms.

Factor Theorem:
According to the Factor Theorem, if $P(x)$ is a polynomial and $a$ is a root, then $(x - a)$ is a factor of the polynomial.

Remainder Theorem:
The Remainder Theorem states that when a polynomial $P(x)$ is divided by another polynomial $d(x)$, the remainder is what remains after the division.

Main Concepts and Formule Chapter 2 Class 9 Mathematics

Algebraic identities -
$(x+y)^{2}=x^{2}+2 x y+y^{2}$
$(x-y)^{2}=x^{2}-2 x y+y^{2}$
$x^{2}-y^{2}=(x+y)(x-y)$
$(x+a)(x+b)=x^{2}+(a+b) x+a b$
$(x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2 x y+2 y z+2 z x$
$(x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}=x^{3}+y^{3}+3 x y(x+y)$
$(x-y)^{3}=x^{3}-3 x^{2} y+3 x y^{2}-y^{3}=x^{3}-y^{3}-3 x y(x-y)$
$x^{3}+y^{3}=(x+y)\left(x^{2}-x y+y^{2}\right)$
$x^{3}-y^{3}=(x-y)\left(x^{2}+x y+y^{2}\right)$
$x^{3}+y^{3}+z^{3}-3 x y z=(x+y+z)\left(x^{2}+y^{2}+z^{2}-x y-y z-z x\right)$

Algebraic expressions are formed using constants, variables, and operators. They can represent a polynomial and can undergo operations like addition, subtraction, multiplication, and division.

Conclusion

The Chapter 2: Polynomials of Class 9 Maths strengthens the foundational understanding of polynomials. It includes essential concepts and questions that are useful for exams.