**Exercise 6.1**

**Question 1: Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer:**

**(i) 3x**

^{2}– 4x + 15**(ii) y**

^{2}+ 2√3**(iii) 3√x + √2 x**

**(iv) x – 4/x**

**(v) x**

^{12}+ y^{3}+ t^{50 }^{ }

**Solution:**

^{2}– 4x + 15 is an expression having only non-negative integral powers of x. So it is a polynomial.

^{2}+ 2√3 is an expression having only non-negative integral power of y. So it is a polynomial.

^{(1/2)}has, rational power of x. So, it is not a polynomial.

^{-1}has, which is not a positive term. So, it is not a polynomial.

^{12}+ y

^{3}+ t

^{50}is an expression having only non-negative integral powers of its variables which is x, y and t. So it is a polynomial.

**Question 2: Write the coefficient of x**

^{2}in each of the following:**(i) 17 – 2x + 7x**

^{2}**(ii) 9 – 12x + x**

^{3}**(iii) π/6 x**

^{2}– 3x + 4**(iv) √3 x – 7**

**Solution:**

^{2}the coefficient of x

^{2}is 7

^{3}the coefficient of x

^{2}is 0

^{2}– 3x + 4 the coefficient of x

^{2}is π/6

^{2}is

**Question 3: Write the degrees of each of the following polynomials:**

**(i) 7x**

^{3}+ 4x^{2}– 3x + 12**(ii) 12 – x + 2x**

^{3}**(iii) 5y – √2**

**(iv) 7**

**(v) 0**

**Solution:**

^{3}+ 4x

^{2}– 3x + 12, Degree is the highest power in the polynomial. So the Degree of the given polynomial is 3.

^{3}Degree is the highest power in the polynomial. So the Degree of the given polynomial is 3.

**Question 4: Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials:**

**(i) x + x**

^{2}+ 4**(ii) 3x – 2**

**(iii) 2x + x**

^{2}**(iv) 3y**

**(v) t**

^{2}+ 1**(vi) 7t**

^{4}+ 4t^{3}+ 3t – 2**Solution:**

^{2}+ 4

^{2}

^{2}+ 1

^{4}+ 4t

^{3}+ 3t – 2

**Exercise 6.2**

**Question 1: If f(x)=2x**

^{3}– 13x**+17x+ 12, find**

**(i) f(2)**

**(ii) f(-3)**

**(iii) f(0)**

**Solution:**

^{3}– 13x

^{3}+17x+ 12

^{3}– 13(2)

^{2}+ 17(2) + 12

^{3}– 13(-3)

^{2}+ 17 x (-3) + 12

^{3}– 13(0)

^{2}+ 17 x 0 + 12

**Question 2: Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following cases:**

**(i) f(x) = 3x + 1, x = −1/3**

**(ii) f(x) = x**

^{2}– 1, x = 1,−1**(iii) g(x) = 3x**

^{2}– 2 , x = 2/√3 , -2/√3**(iv) p(x) = x**

^{3}– 6x^{2}+ 11x – 6 , x = 1, 2, 3**(v) f(x) = 5x – π, x = 4/5**

**(vi) f(x) = x**

^{2}, x = 0**(vii) f(x) = lx + m, x = −m/l**

**(viii) f(x) = 2x + 1, x = 1/2**

**Solution:**

^{2}– 1

^{2}– 1 ………………….(1)

^{2}– 1

^{2}– 1

^{2}– 1.

^{2}– 2

^{2}– 2………………….(1)

^{2}– 2

^{2}–2

^{2}– 2.

^{3}– 6x

^{2}+ 11x – 6

^{3}– 6(1)

^{2}+ 11x 1 – 6

^{3}– 6(2)

^{2}+ 11×2 – 6

^{3}– 6(3)

^{2}+ 11×3 – 6

^{3}– 6x

^{2}+ 11x – 6.

^{2}, x = 0

^{2 }

^{2}.

**Exercise 6.3**

**In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the by actual division: (1 – 8)**

**Question 1: f(x) = x**

^{3}+ 4x^{2}– 3x + 10, g(x) = x + 4**Solution:**

^{3}+ 4x

^{2}– 3x + 10

^{3}+ 4(-4)

^{2}– 3(-4) + 10

**Question 2: f(x) = 4x**

^{4}– 3x^{3}– 2x^{2}+ x – 7, g(x) = x – 1**Solution:**

^{4}–3x

^{3}–2x

^{2}+x–7

^{4}– 3(1)

^{3}– 2(1)

^{2}+ (1) – 7

**Question 3: f(x) = 2x**

^{4}– 6x^{3}+ 2x^{2}– x + 2, g(x) = x + 2**Solution:**

^{4}– 6x

^{3}+ 2x

^{2}– x + 2

^{4}– 6(-2)

^{3}+ 2(-2)

^{2}– (-2) + 2

**Question 4: f(x) = 4x**

^{3}– 12x^{2}+ 14x – 3, g(x) = 2x – 1**Solution:**

^{3}– 12x

^{2}+ 14x – 3

^{3}– 12(1/2)

^{2}+ 14(1/2)–3

**Question 5: f(x) = x**

^{3}– 6x^{2}+ 2x – 4, g(x) = 1 – 2x**Solution:**

^{3}– 6x

^{2}+ 2x – 4

^{3}– 6(1/2)

^{2}+ 2(1/2) – 4

**Question 6: f(x) = x**

^{4}– 3x^{2}+ 4, g(x) = x – 2**Solution:**

^{4}– 3x

^{2}+ 4

^{4}– 3(2)

^{2}+ 4

**Question 7: f(x) = 9x**

^{3}– 3x^{2}+ x – 5, g(x) = x – 2/3**Solution:**

^{3}– 3x

^{2}+ x – 5, g(x) = x – 2/3

^{3}– 3(2/3)

^{2}+ (2/3) – 5

**Exercise 6.4**

**In each of the following, use factor theorem to find whether polynomial g(x) is a factor of polynomial f(x) or, not: (1-7)**

**Question 1: f(x) = x**

^{3}– 6x^{2}+ 11x – 6; g(x) = x – 3**Solution:**

^{3}– 6x

^{2}+ 11x – 6

^{3}– 6(3)

^{2}+11 x 3 – 6

**Question 2: f(x) = 3x**

^{4}+ 17x^{3}+ 9x^{2}– 7x – 10; g(x) = x + 5**Solution:**

^{4}+ 17x

^{3}+ 9x

^{2}– 7x – 10

^{4}+ 17(-5)

^{3}+ 9(-5)

^{2}– 7(-5) – 10

**Question 3: f(x) = x**

^{5}+ 3x^{4}– x^{3}– 3x^{2}+ 5x + 15, g(x) = x + 3**Solution:**

^{5}+ 3(-3)

^{4}– (-3)

^{3}– 3(-3)

^{2}+ 5(-3) + 15

**Question 4: f(x) = x**

^{3}– 6x^{2}– 19x + 84, g(x) = x – 7**Solution:**

^{3}– 6x

^{2}– 19x + 84

^{3}– 6(7)

^{2}– 19 x 7 + 84

**Question 5: f(x) = 3x**

^{3}+ x^{2}– 20x + 12, g(x) = 3x – 2**Solution:**

^{3}+ x

^{2}– 20x + 12

^{3}+ (2/3)

^{2}– 20(2/3) + 12

**Question 6: f(x) = 2x**

^{3}– 9x^{2}+ x + 12, g(x) = 3 – 2x**Solution:**

^{3}– 9x

^{2}+ x + 12

^{3}– 9(3/2)

^{2}+ (3/2) + 12

**Question 7: f(x) = x**

^{3}– 6x^{2}+ 11x – 6, g(x) = x^{2}– 3x + 2**Solution:**

^{3}– 6x

^{2}+ 11x – 6

^{2}– 3x + 2 = 0

^{2}– x – 2x + 2 = 0

^{3}– 6(1)

^{2}+ 11(1) – 6

^{3}– 6(2)

^{2}+ 11(2) – 6

**Question 8: Show that (x – 2), (x + 3) and (x – 4) are factors of x**

^{3}– 3x^{2}– 10x + 24.**Solution:**

^{3}– 3x

^{2}– 10x + 24

^{3}– 3(2)

^{2}– 10 x 2 + 24

^{3}– 3(-3)

^{2}– 10 (-3) + 24

^{3}– 3(4)

^{2}– 10 x 4 + 24

**Question 9: Show that (x + 4), (x – 3) and (x – 7) are factors of x3 – 6x2 – 19x + 84.**

**Solution:**

^{3}– 6x

^{2}– 19x + 84

^{3}– 6(-4)

^{2}– 19(-4) + 84

^{3}– 6(3)

^{2}– 19 x 3 + 84

^{3}– 6(7)

^{2}– 19 x 7 + 84

**Exercise 6.5**

**Using factor theorem, factorize each of the following polynomials:**

**Question 1: x**

^{3}+ 6x^{2}+ 11x + 6**Solution:**

^{3}+ 6x

^{2}+ 11x + 6

^{3}+ 6(−1)

^{2}+ 11(−1) + 6

^{3}+ 6(−2)

^{2}+ 11(−2) + 6

^{3}+ 6(−3)

^{2}+ 11(−3) + 6

**Question 2: x**

^{3}+ 2x^{2}– x – 2**Solution:**

^{3}+ 2x

^{2}– x – 2

^{3}+ 2(1)

^{2}– 1 – 2

^{3}+ 2(-1)

^{2}– 1 – 2

^{3}+ 2(-2)

^{2}– (-2) – 2

^{3}+ 2(2)

^{2}– 2 – 2

**Question 3: x**

^{3}– 6x^{2}+ 3x + 10**Solution:**

^{3}– 6x

^{2}+ 3x + 10

^{3}– 6(-1)

^{2}+ 3(-1) + 10

^{3}– 6(-2)

^{2}+ 3(-2) + 10

^{3}– 6(2)

^{2}+ 3(2) + 10

^{3}– 6(5)

^{2}+ 3(5) + 10

**Question 4: x**

^{4}– 7x^{3}+ 9x^{2}+ 7x- 10**Solution:**

^{4}– 7x

^{3}+ 9x

^{2}+ 7x- 10

^{4}– 7(1)

^{3}+ 9(1)

^{2}+ 7(1) – 10

^{4}– 7(-1)

^{3}+ 9(-1)

^{2}+ 7(-1) – 10

^{4}– 7(2)

^{3}+ 9(2)

^{2}+ 7(2) – 10

^{4}– 7(5)

^{3}+ 9(5)

^{2}+ 7(5) – 10

**Question 5: x**

^{4}– 2x^{3}– 7x^{2}+ 8x + 12**Solution:**

^{4}– 2x

^{3}– 7x

^{2}+ 8x + 12

^{4}– 2(1)

^{3}– 7(1)

^{2}+ 8(1) + 12

^{4}– 2(-1)

^{3}– 7(-1)

^{2}+ 8(-1) + 12

^{4}– 2(-2)

^{3}– 7(-2)

^{2}+ 8(-2) + 12

^{4}– 2(2)

^{3}– 7(2)

^{2}+ 8(2) + 12

^{4}– 2(3)

^{3}– 7(3)

^{2}+ 8(3) + 12

**Question 6: x**

^{4}+ 10x^{3}+ 35x^{2}+ 50x + 24**Solution:**

^{4}+ 10x

^{3}+ 35x

^{2}+ 50x + 24

^{4}+ 10(-1)

^{3}+ 35(-1)

^{2}+ 50(-1) + 24

^{4}+ 10(-2)

^{3}+ 35(-2)

^{2}+ 50(-2) + 24

^{4}+ 10(-3)

^{3}+ 35(-3)

^{2}+ 50(-3) + 24

^{4}+ 10(-4)

^{3}+ 35(-4)

^{2}+ 50(-4) + 24

^{4}+ 10x

^{3}+ 35x

^{2}+ 50x + 24.

**Question 7: 2x**

^{4}– 7x^{3}– 13x^{2}+ 63x – 45**Solution:**

^{4}– 7x

^{3}– 13x

^{2}+ 63x – 45

^{4}is 2.

^{4}– 7(1)

^{3}– 13(1)

^{2}+ 63(1) – 45

^{4}– 7(3)

^{3}– 13(3)

^{2}+ 63(3) – 45

^{2}- 3x - 1x + 3

^{2}- 4x + 3

^{2}- 4x + 3 is root of 2x

^{4}– 7x

^{3}– 13x

^{2}+ 63x – 45.

^{2}+ x – 15

^{2}+ 6x – 5x – 15

^{4}– 7x

^{3}– 13x

^{2}+ 63x – 45 are (x – 1) (x – 3) (2x + 5) (x – 3).

**Exercise VSAQs : ->**

**Question 1: Define zero or root of a polynomial**

**Solution:**

**Question 2: If x = 1/2 is a zero of the polynomial f(x) = 8x**

^{3}+ ax^{2}– 4x + 2, find the value of a.**Solution:**

^{3}+ ax

^{2}– 4x + 2

^{3}+ a(1/2)

^{2}– 4(1/2) + 2 = 0

**Question 3: Write the remainder when the polynomial f(x) = x**

^{3}+ x^{2}– 3x + 2 is divided by x + 1.**Solution:**

^{3}+ x

^{2}– 3x + 2

^{3}+ (-1)

^{2}– 3(-1) + 2

**Question 4: Find the remainder when x**

^{3}+ 4x^{2}+ 4x - 3 if divided by x**Solution:**

^{3}+ 4x

^{2}+ 4x - 3

^{3}+ 4x

^{2}+ 4x - 3

^{3}+ 4(0)

^{2}+ 4(0) -3

**Question 5: If x + 1 is a factor of x**

^{3}+ a, then write the value of a.**Solution:**

^{3}+ a

^{3}+ a

^{3}+ a = 0

**Question 6: If f(x) = x**

^{4}– 2x^{3}+ 3x^{2}– ax – b when divided by x – 1, the remainder is 6, then find the value of a + b.**Solution:**

^{4}– 2x

^{3}+ 3x

^{2}– ax – b

^{4}– 2(1)

^{3}+ 3(1)

^{2}– a(1) – b = 6