NCERT Solutions for Chapter 2 Polynomials Class 9 Maths
Chapter Name | NCERT Solutions for Chapter 2 Polynomials |
Class | Class 9 |
Topics Covered |
|
Related Study Materials |
|
Short Revision for Ch 2 Polynomials Class 9 Maths
- In the polynomial p(x), the highest exponent of x is called the degree of the polynomial p(x).
- Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomials respectively.
- A linear polynomial in x is of the form ax + b, a ≠ 0.
- A quadratic polynomial in x is of the form ax2 + bx + c, a ≠ 0.
- A cubic polynomial in x is of the form ax3 + bx2 + cx + d, a ≠ 0.
- A constant polynomial is free from a variable. 0, 7, 8, -4, etc. are examples of constant polynomials.
- 0 is also called the zero polynomial.
- The degree of a non - zero constant polynomial is 0.
- The degree of the zero polynomial does not exist.
- In the polynomial 3x3 - 4x + 7, the expressions 3x3 , - 4x and 7 are called the terms of the polynomial.
- If a polynomial consists of only one variable, then the polynomial is called polynomial in one variable.
- Polynomials consisting of one term, two terms and three terms are called monomial, binomial and trinomial respectively.
- A real number k is a zero of the polynomial p(x), if p(k) = 0.
- Every linear polynomial in one variable has a unique zero.
- A non - zero constant polynomial has no zero.
- Every real number is a zero of the zero polynomial.
- A quadratic polynomial has at most two zeroes.
- A cubic polynomial has at most three zeroes.
- The polynomial equation of the polynomial p(x) is given by p(x) = 0.
- Remainder Theorem : If p(x) is a polynomial of degree greater than or equal to 1 and f(x) is divided by the linear polynomial x - a, then the remainder is f(a).
- Dividend = (Divisor × Quotient ) + Remainder.
- If dividend, divisor, quotient and remainder are respectively f(x), g(x), q(x) and r(x) such that degree of f(x) ≥ degree of g(x) with g(x) ≠ 0, then f(x) = g(x) × q(x) + r(x) where r(x) = 0 or degree of r(x) < degree of g(x).
- Factor Theorem : If x -a is a factor of f(x). then f(a) = 0.
- (x + y)2 = x2 + 2xy + y2
- (x − y)2 = x2 − 2xy + y2
- x2 – y2 = (x + y)(x – y)
- (x + a)(a + b) = x2 + (a + b)x + ab
- (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
- (x + y)3 = x3 + y3 + 3xy(x + y) = x3 + 3x2y + 3xy2 + y3
- (x – y)3 = x3 – y3 – 3xy(x – y) = x3 – 3x2y + 3xy2 – y3
- x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)
- x3 + y3 + z3 = 3xyz, if x + y + x = 0
Exercise 2.1
1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) 4x2 – 3x + 7
(ii) y2 + √2
(iii) 3√t + t√2
(iv) y + 2/y
(v) x10 + y3 + t50
(i) 2 + x2 +x
(ii) 2 - x2 + x3
(iv) √2x - 1
3. Give one example each of a binomial of degree 35, and of a monomial of degree 100.
4. Write the degree of each of the following polynomials:
(i) 5x3+4x2+7x
(ii) 4 - y2
(iii) 5t - √7
(iv) 3
5. Classify the following as linear, quadratic and cubic polynomials:
(i) x2 + x
(ii) x – x3
(iii) y + y2 + 4
(iv) 1 + x
(v) 3t
(vi) r2
(vii) 7x3
Exercise 2.2
(i) x = 0
(ii) x = – 1
(iii) x = 2
(i) p(y) = y2 − y + 1
(ii) p(t) = 2 + t + 2t2 – t3
(iii) p(x) = x3
(iv) p(x) = (x – 1)(x+ 1)
3. Verify whether the following are zeroes of the polynomial, indicated against them.
(i) p(x) = 3x +1, x = −1/3
(ii) p(x) = 5x – Ï€, x = 4/5
(iii) p(x) = x2 – 1, x = 1, −1
(iv) p(x) = (x + 1)(x – 2), x = -1, 2
(v) p(x) = x3 , x = 0
(vi) p(x) = lx + m, x = -m/l
(vii) p(x) = 3x2 – 1, x = -1/√3, 2/√3
(viii) p(x) = 2x + 1, x = 1/2
4. Find the zero of the polynomials in each of the following cases:
(i) p(x) = x+5
(ii) p(x) = x - 5
(iii) p(x) = 2x + 5
(iv) p(x) = 3x - 2
(v) p(x) = 3x
(vi) p(x) = ax, a ≠ 0
(vii) p(x) = cx + d, c ≠ 0, c, d are real numbers.
Exercise 2.3
(i) x + 1
(ii) x -1/2
(iii) x
(iv) x + π
(v) 5 + 2x
Exercise 2.4
1. Determine which of the following polynomials has (x + 1) a factor:
(i) x3+ x2 + x + 1
(ii) x4 + x3 + x2 + x + 1
(iii) x4+ 3x3 + 3x2 + x + 1
(iv) x3 – x2– (2+√2)x +√2
2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
(i) p(x) = 2x3+x2–2x–1, g(x) = x+1
(ii) p(x)=x3+3x2+3x+1, g(x) = x+2
(iii) p(x)=x3–4x2+x+6, g(x) = x–3
3. Find the value of k, if x–1 is a factor of p(x) in each of the following cases:
(i) p(x) = x2+x+k
(ii) p(x) = 2x2+kx+√2
(iii) p(x) = kx2–√2x + 1
(iv) p(x) = kx2 – 3x + k
4. Factorize:
(i) 12x2–7x+1
(ii) 2x2+7x+3
(iii) 6x2+5x-6
(iv) 3x2–x–4
5. Factorize:
(i) x3–2x2–x +2
(ii) x3–3x2–9x–5
(iii) x3+13x2+32x+20
(iv) 2y3+y2–2y–1
Exercise 2.5
1. Use suitable identities to find the following products:
(i) (x+4)(x +10)
(ii) (x+8)(x –10)
(iii) (3x+4)(3x–5)
(iv) (y2+3/2)(y2-3/2)
(v) (3 - 2x)(3 + 2x).
2. Evaluate the following products without multiplying directly:
(i) 103×107
(ii) 95×96
(iii) 104×96
3. Factorize the following using appropriate identities:
(i) 9x2+6xy+y2
(ii) 4y2−4y+1
(iii) x2–y2/100
4. Expand each of the following, using suitable identities:
(i) (x+2y+4z)2
(ii) (2x−y+z)2
(iii) (−2x+3y+2z)2
(iv) (3a –7b–c)2
(v) (–2x+5y–3z)2
(vi) (1/4)a-(1/2)b +1)2
5. Factorize:
(i) 4x2+9y2+16z2+12xy–24yz–16xz
(ii ) 2x2+y2+8z2–2√2xy+4√2yz–8xz
6. Write the following cubes in expanded form:
(i) (2x+1)3
(ii) (2a−3b)3
(iii) ((3/2)x+1)3
(iv) (x−(2/3)y)3
7. Evaluate the following using suitable identities:
(i) (99)3
(ii) (102)3
(iii) (998)3
8. Factorise each of the following:
(i) 8a3+b3+12a2b+6ab2
(ii) 8a3–b3–12a2b+6ab2
(iii) 27–125a3–135a +225a2
(iv) 64a3–27b3–144a2b+108ab2
(v) 27p3–(1/216)−(9/2) p2+(1/4)p
9. Verify:
(i) x3+y3 = (x+y)(x2–xy+y2)
(ii) x3–y3 = (x–y)(x2+xy+y2)
10. Factorize each of the following:
(i) 27y3+125z3
(ii) 64m3–343n3
11. Factorise: 27x3+y3+z3– 9xyz.
12. Verify that:
x3+y3+z3–3xyz = (1/2) (x+y+z)[(x–y)2+(y–z)2+(z–x)2]
13. If x+y+z = 0, show that x3+y3+z3 = 3xyz.
14. Without actually calculating the cubes, find the value of each of the following:
(i) (−12)3+(7)3+(5)3
(ii) (28)3+(−15)3+(−13)3
15. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
(i) Area : 25a2–35a+12
(ii) Area : 35y2+13y–12
16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
(i) Volume : 3x2 – 12x
(ii) Volume : 12ky2 + 8ky – 20k
