2. Polynomials

Exercise 2.1

1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
   1. 4x² – 3x + 7
   2. y² + 2
   3. 3/2 t + t
   4. y + √2y
   5. x¹⁰ + y³ + t⁵⁰
2. Write the coefficients of x² in each of the following:
   1. 2 + x² + x
   2. 2 – x² + x³
   3. 2πx + x²
   4. 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:
   1. 5x³ + 4x² + 7x
   2. 4 – y²
   3. 5t – 7
   4. 3
5. Classify the following as linear, quadratic and cubic polynomials:
   1. x² + x
   2. x – x³
   3. y + y² + 4
   4. 1 + x
   5. 3t
   6. r²
   7. 7x

2.3 Zeroes of a Polynomial

Exercise 2.2

1. Find the value of the polynomial 5x – 4x² + 3 at:
   1. x = 0
   2. x = –1
   3. x = 2
2. Find p(0), p(1) and p(2) for each of the following polynomials:
   1. p(y) = y² – y + 1
   2. p(t) = 2 + t + 2t² – t³
   3. p(x) = x³
   4. p(x) = (x – 1)(x + 1)
3. Verify whether the following are zeroes of the polynomial, indicated against them:
   1. p(x) = 3x + 1, x = –1/3
   2. p(x) = 5x – π, x = 4/5
   3. p(x) = x² – 1, x = 1, –1
   4. p(x) = (x + 1)(x – 2), x = –1, 2
   5. p(x) = x², x = 0
   6. p(x) = lx + m, x = –m/l
   7. p(x) = 3x² – 1, x = 1/√2, –1/√3
   8. p(x) = 2x + 1, x = –1/2
4. Find the zero of the polynomial in each of the following cases:
   1. p(x) = x + 5
   2. p(x) = x – 5
   3. p(x) = 2x + 5
   4. p(x) = 3x – 2
   5. p(x) = 3x
   6. p(x) = ax, a ≠ 0
   7. p(x) = cx + d, c ≠ 0; c, d ∈ ℝ

2.4 Factorisation of Polynomials

Exercise 2.3

1. Determine which of the following polynomials has (x + 1) as a factor:
   1. x³ + x² + x + 1
   2. x⁴ + x³ + x² + x + 1
   3. x⁴ + 3x³ + 3x² + x + 1
   4. x³ – x² – (2 + 2 + 2)x
2. Use the Factor Theorem to determine whether g(x) is a factor of p(x) in each of the following cases:
   1. p(x) = 2x³ + x² – 2x – 1, g(x) = x + 1
   2. p(x) = x³ + 3x² + 3x + 1, g(x) = x + 2
   3. 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:
   1. p(x) = x² + x + k
   2. p(x) = 2x² + kx + 2
   3. p(x) = kx² – 2x + 1
   4. p(x) = kx² – 3x + k
4. Factorise:
   1. 12x² – 7x + 1
   2. 2x² + 7x + 3
   3. 6x² + 5x – 6
   4. 3x² – x – 4
5. Factorise:
   1. x³ – 2x² – x + 2
   2. x³ – 3x² – 9x – 5
   3. x³ + 13x² + 32x + 20
   4. 2y³ + y² – 2y – 1

2.5 Algebraic Identities

Exercise 2.4

1. Use suitable identities to find the following products:
   1. (x + 4)(x + 10)
   2. (x + 8)(x – 10)
   3. (3x + 4)(3x – 5)
   4. (y² + √3/2)(y² – √3/2)
   5. (3 – 2x)(3 + 2x)
2. Evaluate the following products without multiplying directly:
   1. 103 × 107
   2. 95 × 96
   3. 104 × 96
3. Factorise the following using appropriate identities:
   1. 9x² + 6xy + y²
   2. 4y² – 4y + 1
   3. x² – (2/100)y
4. Expand each of the following using suitable identities:
   1. (x + 2y + 4z)²
   2. (2x – y + z)²
   3. (–2x + 3y + 2z)²
   4. (3a – 7b – c)²
   5. (–2x + 5y – 3z)²
   6. [(1/2)a – (1/4)b + 1]²
5. Factorise:
   1. 4x² + 9y² + 16z² + 12xy – 24yz – 16xz
   2. 2x² + y² + 8z² – 2√2 xy + 4√2 yz – 8xz
6. Write the following cubes in expanded form:
   1. (2x + 1)³
   2. (2a – 3b)³
   3. [(3/2)x + 1]³
   4. [(2/3)x – y]³
7. Evaluate the following using suitable identities:
   1. (99)³
   2. (102)³
   3. (998)³
8. Factorise each of the following:
   1. 8a³ + b³ + 12a²b + 6ab²
   2. 8a³ – b³ – 12a²b + 6ab²
   3. 27 – 125a³ – 135a + 225a²
   4. 64a³ – 27b³ – 144a²b + 108ab²
   5. 27p³ – 1/216 – (9/2)p + (1/4)p²
9. Verify:
   1. x³ + y³ = (x + y)(x² – xy + y²)
   2. x³ – y³ = (x – y)(x² + xy + y²)
10. Factorise:
    1. 27y³ + 125z³
    2. 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:
    1. (–12)³ + 7³ + 5³
    2. 28³ + (–15)³ + (–13)³
15. Give possible expressions for the length and breadth of each of the following rectangles, where areas are:
    1. Area: 25a² – 35a + 12
    2. Area: 35y² + 13y – 12
16. What are the possible expressions for the dimensions of the cuboids whose volumes are given below?
    1. Volume: 3x² – 12x
    2. Volume: 12ky² + 8ky – 20k

2.6 Summary