CBSE Worksheet for chapter-4 Polynomial class 9

Worksheet For class 9

Find CBSE Worksheet for chapter-3 Polynomial class 9

CLASS-9

BOARD: CBSE

Mathematic Worksheet - 4

TOPIC: Polynomial

For other CBSE Worksheet for class 9 Mathematic check out main page of Physics Wallah.

 

Summary

An algebraic expression f(x) of the form f(x)= a₀ + a₁x +a₂x²+…….+a_n xⁿ, where a₀,a_1, a₂……. a_n are real numbers and all the index of x are non negative integers is called polynomial in x and the highest index n is called as the degree of polynomial.

Important Formulas

• (x+a)² =x² +2ax+a²
• (x-a)² =x² – 2ax +a²
• x²-a² =(x+a) (x-a)
• x³+a³=(x+a) (x²-ax+a²)=(x+a)³ –3xa(x+a)
• x³-a³=(x-a) (x²+ax+a²)=(x+a)³ +3xa(x-a)
• (a + b + c) ²=a²+b²+c²+2ab+2bc+2ac
• (a + b) ³= a³+b³+3ab (a + b)
• (a-b) ³= a³-b³-3ab (a-b)
• a³+b³+c³-3abc = (a +b +c) (a²+b²+c²-ab-bc-ca)
• SPECIAL CASE : If a+ b +c=0 then a³+b³+c³=3abc

Zeros or Roots of a Polynomial

A real number ‘a’ is a zero of a polynomial f(x),if f(a)=0.Here ‘a’ is called a root of the equation f(x) = 0

Factor Theorem

Let p(x) be a polynomial of degree greater than or equal to ‘a’, be a real number such that p(a)=0,then (x-a) is a factor of p(x),then p(a) = 0.

Remainder Theorem

Let p(x) be any polynomial of degree greater than or equal to one and ‘a’ be any real number. If p(x)is a dividend by (x-a),then remainder is equal to p(a).let q(x) be the quotient and r(x) be the remainder when p(x) is divided by (x-a),then

Dividend=Divisor x Quotient + Remainder

Objective

Q1. Factors of (42-x-x²) are

1. (x-7) (x-6)
2. (x+7) (6-x)
3. (x+7) (6-x)
4. (x+7) (x+6)

Q2. Factors of are(x²+x/6-1/6)

1. 1/6 (2x+1)(3x+1)
2. 1/6 (2x+1)(3x-1)
3. 1/6 (2x-1)(3x-1)
4. 1/6 (2x-1)(3x+1)

Q3. Factors of polynomial are

1. (x-2) (x+3) (x-4)
2. (x-+) (x+3) (x+4)
3. (x+2) (x-3) (x-4)
4. (x-2) (x-3) (x-4)

Q4. If (x+a) is a factor of x²+pq+q and x²+mx+n then the value of a is

1. (m-p)/(n-q)
2. (n-p)/(m-q)
3. (n+p)/(m+q)
4. (m+p)/(n+q)

Subjective

1. Factorize x⁴+x³+7x²-x+6.
2. Determine whether x-3 is a factor of polynomial p(x)= x³ 3x² +4x-12
3. If p(x)= 4x⁴ +5x³ -12x²-11x+5,g(x)=4x+5
4. Determine if (x+1) is a factor of (2-√2)x + √2
5. Factorize the following polynomials:
   1. x³ -23x²+142x-120
   2. 2y³ +y²-2y-1
   3. x⁴ -5x²+4
   4. x³ -10x²-53x-42

​Solutions

Objective

1. 3
2. 2
3. 1
4. 2

Subjective

1. (x+1)(x-1)(x+3)(x-2)
2. yes
3. (x+1) is a factor
4. (x-1)(x-10)(x-12)

For other CBSE Worksheet for class 9 Mathematics check out main page of Physics Wallah.