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Chapters
1: Real Numbers
Algebra
▶ 2: Polynomials
3: Linear Equations in Two Variables
4: Quadratic Equations
5: Arithmetic Progression
Coordinate Geometry
6: Coordinate Geometry
Geometry
7: Triangles
8: Circles
9: Constructions
Trigonometry
10: Trignometric Ratios
11: T-Ratios of Some Particular Angles
12: Trigonometric Ratios of Some Complemantary Angles
13: Trigonometric identities
14: Heights and Distances
Mensuration
15: Perimeter And Area of Plane Figures
16: Area of Circle, Sector and Segment
17: Volumes and Surface Areas of Solids
Statistics and Probability
18: Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive
19: Probability
Chapter 20: Additional Questions
![R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 2 - Polynomials - Shaalaa.com](/images/mathematics-english-class-10_6:8f062ea57bdf49abb4f6d22550b39d56.jpg "R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 2 - Polynomials")
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Solutions for Chapter 2: Polynomials
Below listed, you can find solutions for Chapter 2 of CBSE, Karnataka Board R.S. Aggarwal for Mathematics [English] Class 10.
R.S. Aggarwal solutions for Mathematics [English] Class 10 2 Polynomials EXERCISE 2A [Pages 52 - 53]
Find the zeroes of the quadratic polynomial f(x) = x2 + 3x – 10 and verify the relation between its zeroes and coefficients.
Find the zeroes of the polynomial f(x) = x2 – 2x – 8 and verify the relation between its zeroes and coefficients.
Find the zeros of the polynomial f(x) = x2 + 7x + 12 and verify the relation between its zeroes and coefficients.
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
2x2 – x – 6
Find the zeroes of the quadratic polynomial f(x) = 4x2 – 4x – 3 and verify the relation between its zeroes and coefficients.
Find the zeroes of the quadratic polynomial f(x) = 5x2 – 4 – 8x and verify the relationship between the zeroes and coefficients of the given polynomial.
Find the zeroes of the quadratic polynomial 2x2 – 11x + 15 and verify the relation between the zeroes and the coefficients.
Find the zeroes of the polynomial f(x) = `2sqrt(3)x^2 - 5x + sqrt(3)` and verify the relation between its zeroes and coefficients.
Find the zeroes of the quadratic polynomial 4x2 – 4x + 1 and verify the relation between the zeroes and the coefficients.
Find the zeroes of the quadratic polynomial (3x2 – x – 4) and verify the relation between the zeroes and the coefficients.
Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:
5x2 + 10x
Find the zeroes of the quadratic polynomial (8x2 – 4) and verify the relation between the zeroes and the coefficients.
If α and β are the zeros of the polynomial p(x) = 2x2 + 5x + k satisfying the relation α2 + β2 + αβ = `21/4` then find the value of k.
Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.
Find the quadratic polynomial, the sum of whose zeros is 0 and their product is –1. Hence, find the zeros of the polynomial.
Find the quadratic polynomial, the sum of whose zeroes is `(5/2)` and their product is 1. Hence, find the zeros of the polynomial.
Find the quadratic polynomial whose zeros are 2 and –6. Verify the relation between the coefficients and the zeros of the polynomial.
Find the quadratic polynomial whose zeroes are `2/3` and `(-1)/4`. Verify the relation between the coefficients and the zeroes of the polynomial.
If (x + a) is a factor of (2x2 + 2ax + 5x + 10), then find the value of a.
If `2/3` and –3 are the zeros of the quadratic polynomial ax2 + 7x + b then find the values of a and b.
If the sum of the squares of zeros of the polynomial f(x) = x2 – 8x + k is 40, find the value of k.
R.S. Aggarwal solutions for Mathematics [English] Class 10 2 Polynomials EXERCISE 2B [Pages 63 - 64]
Verify that 3, –2, 1 are the zeros of the cubic polynomial p(x) = (x3 – 2x2 – 5x + 6) and verify the relation between it zeros and coefficients.
Verify that 5, –2 and `1/3` are the zeroes of the cubic polynomial p(x) = (3x3 – 10x2 – 27x + 10) and verify the relation between its zeros and coefficients.
Find a cubic polynomial whose zeroes are 2, –3 and 4.
Find a cubic polynomial whose zeroes are `1/2`, 1 and –3.
Find a cubic polynomial with the sum of its zeroes, sum of the products of its zeroes taken two at a time and the product of its zeroes as 5, –2 and –24 respectively.
Find the quotient and the remainder when f(x) = x3 – 3x2 + 5x – 3 is divided by g(x) = x2 – 2.
Find the quotient and the remainder when f(x) = x4 – 3x2 + 4x + 5 is divided by g(x) = x2 – x + 1.
Find the quotient and the remainder when f(x) = x4 – 5x + 6 is divided by g(x) = 2 – x2.
By actual division, show that x2 – 3 is a factor of 2x4 + 3x3 – 2x2 – 9x – 12.
If the polynomial (x4 + 2x3 + 8x2 + 12x + 18) is divided by another polynomial (x2 + 5), the remainder comes out to be (px + q). Find the values of p and q.
On dividing 3x3 + x2 + 2x + 5 is divided by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).
Verify division algorithm for the polynomial f(x)= (8 + 20x + x2 – 6x3) by g(x) = (2 + 5x – 3x2).
It is given that –1 is one of the zeroes of the polynomial x3 + 2x2 – 11x – 12. Find all the zeroes of the given polynomial.
If 1 and –2 are two zeroes of the polynomial (x3 – 4x2 – 7x + 10), find its third zero.
If 3 and –3 are two zeroes of the polynomial (x4 + x3 – 11x2 – 9x + 18), find all the zeroes of the given polynomial.
If 2 and –2 are two zeros of the polynomial 2x4 – 5x3 – 11x2 + 20x + 12, find all the zeros of the given polynomial.
Find all the zeroes of (x4 + x3 – 23x2 – 3x + 60), if it is given that two of its zeroes are `sqrt(3)` and `-sqrt(3)`.
Obtain all other zeroes of (x4 + 4x3 – 2x2 – 20x – 15) if two of its zeroes are `sqrt(5)` and `-sqrt(5)`.
Obtain all the zeros of the polynomial x4 + x3 – 14x2 – 2x + 24 if two of its zeros are `sqrt(2)` and `-sqrt(2)`.
Find all the zeros of 2x4 – 13x3 + 19x2 + 7x – 3 if two of its zeros are `(2 + sqrt(3))` and `(2 - sqrt(3))`.
One zero of the polynomial 3x3 + 16x2 + 15x – 18 is `2/3`. Find the other zeros of the polynomial.
Find all the zeros of 2x4 – 3x3 – 3x2 + 6x – 2 if it is given that two of its zeros are 1 and `1/2`.
Find all the zeroes of polynomial (2x4 – 11x3 + 7x2 + 13x – 7), it being given that two of its zeroes are `(3 + sqrt(2))` and `(3 - sqrt(2))`.
R.S. Aggarwal solutions for Mathematics [English] Class 10 2 Polynomials EXERCISE 2C [Pages 65 - 67]
Very-Short-Answer Questions
If one zero of the polynomial x2 – 4x + 1 is `(2 + sqrt(3))`, write the other zero.
Find the zeros of the polynomial x2 + x – p(p + 1).
Find the zeros of the polynomial x2 – 3x – m(m + 3).
Find α, β are the zeros of polynomial α + β = 6 and αβ = 4 then write the polynomial.
If one zero of the quadratic polynomial kx2 + 3x + k is 2, then find the value of k.
If 3 is a zero of the polynomial 2x2 + x + k, find the value of k.
If –4 is a zero of the polynomial x2 – x – (2k + 2) is –4, then find the value of k.
If 1 is a zero of the quadratic polynomial ax2 – 3(a – 1)x – 1 is 1, then find the value of a.
If –2 is a zero of the polynomial 3x2 + 4x + 2k, then find the value of k.
Write the zeros of the polynomial f(x) = x2 – x – 6.
If the sum of the zeros of the quadratic polynomial kx2 – 3x + 5 is 1, write the value of k.
If the product of the zeros of the quadratic polynomial x2 – 4x + k is 3, then write the value of k.
If (x + a) is a factor of (2x2 + 2ax + 5x + 10), then find the value of a.
If (a – b), a and (a + b) are zeros of the polynomial 2x3 – 6x2 + 5x – 7, write the value of a.
If x3 + x2 – ax + b is divisible by (x2 – x), write the values of a and b.
If α and β are the zeros of the polynomial 2x2 + 7x + 5, write the value of α + β + αβ.
State division algorithm for polynomials.
Find the sum of the zeros and the product of zeros of a quadratic polynomial are `-1/2` and –3 respectively. Write the polynomial.
Short-Answer Questions
Find the zeroes of the quadratic polynomial f(x) = 6x2 – 3.
Find the zeroes of the quadratic polynomial `f(x) = 4sqrt(3)x^2 + 5x - 2sqrt(3)`.
If α, β are the zeroes of the polynomial f(x) = x2 – 5x + k such that α – β = 1, find the value of k = ?
If α and β are the zeros of the polynomial f(x) = 6x2 + x – 2, find the value of `(α/β + α/β)`.
If α, β are the zeroes of the polynomial f(x) = 5x2 – 7x + 1 then `1/α + 1/β = ?`
If α, β are the zeroes of the polynomial f(x) = x2 + x – 2, then `(α/β - α/β)`.
If the zeroes of the polynomial f(x) = x3 – 3x2 + x + 1 are (a – b), a and (a + b), find the values of a and b.
R.S. Aggarwal solutions for Mathematics [English] Class 10 2 Polynomials MULTIPLE-CHOICE QUESTIONS (MCQ) [Pages 69 - 71]
Choose the correct answer in each of the following questions:
Which of the following is a polynomial?
`x^2 - 5x + 4sqrt(x) + 3`
`x^(3//2) - x + x^(1//2) + 1`
`sqrt(x) + 1/sqrt(x)`
`sqrt(2)x^2 - 3sqrt(3)x + sqrt(6)`
Which of the following is not a polynomial?
`sqrt(3)x^2 - 2sqrt(3)x + 5`
`9x^2 - 4x + sqrt(2)`
`3/2 x^3 + 6x^2 - 1/sqrt(2)x - 8`
`x + 3/x`
The zeros of the polynomial x2 – 2x – 3 are ______.
–3, 1
–3, –1
3, –1
3, 1
The zeros of the polynomial `x^2 - sqrt(2)x - 12` are ______.
`sqrt(2), -sqrt(2)`
`3sqrt(2), -2sqrt(2)`
`-3sqrt(2), 2sqrt(2)`
`3sqrt(2), 2sqrt(2)`
The zeros of the polynomial `4x^2 + 5sqrt(2)x - 3` are ______.
`-3sqrt(2), sqrt(2)`
`-3sqrt(2), sqrt(2)/2`
`(-3sqrt(2))/2, sqrt(2)/4`
none of these
The zeros of the polynomial `x^2 + 1/6x - 2` are ______.
–3, 4
`(-3)/2, 4/3`
`(-4)/3, 3/2`
none of these
The zeros of the polynomial `7x^2 - (11x)/3 - 2/3` are ______.
`2/3, (-1)/7`
`2/7, (-1)/3`
`(-2)/3, (1)/7`
none of these
The sum and the product of the zeros of a quadratic polynomial are 3 and –10 respectively. The quadratic polynomial is ______.
x2 – 3x + 10
x2 + 3x – 10
x2 – 3x – 10
x2 + 3x + 10
A quadratic polynomial whose zeros are 5 and –3, is ______.
x2 + 2x – 15
x2 – 2x + 15
x2 – 2x – 15
none of these
A quadratic polynomial whose zeros are `3/5` and `(-1)/2`, is ______.
10x2 + x + 3
10x2 + x – 3
10x2 – x + 3
10x2 – x – 3
The zeros of the quadratic polynomial x2 + 88x + 125 are ______.
both positive
both negative
one positive and one negative
both equal
If α and β are the zeros of x2 + 5x + 8 then the value of (α + β) is ______.
5
–5
8
–8
If α and β are the zeros of 2x2 + 5x – 9 then the value of αβ is ______.
`(-5)/2`
`5/2`
`(-9)/2`
`9/2`
If one zero of the quadratic polynomial kx2 + 3x + k is 2 then the value of k is ______.
`5/6`
`(-5)/6`
`6/5`
`(-6)/5`
If one zero of the quadratic polynomial (k – 1)x2 + kx + 1 is –4 then the value of k is ______.
`(-5)/4`
`5/4`
`(-4)/3`
`4/3`
If –2 and 3 are the zeros of the quadratic polynomial x2 + (a + 1)x + b then
a = –2, b = 6
a = 2, b = –6
a = –2, b = –6
a = 2, b = 6
If one zero of 3x2 + 8x + k be the reciprocal of the other then k = ?
3
–3
`1/3`
`(-1)/3`
If the sum of the zeros of the quadratic polynomial kx2 + 2x + 3k is equal to the product of its zeros then k = ?
`1/3`
`(-1)/3`
`2/3`
`(-2)/3`
If α, β are the zeros of the polynomial x2 + 6x + 2 then `(1/α + 1/β) = ?`
3
–3
12
–12
If α, β, γ are the zeros of the polynomial x3 – 6x2 – x + 30 then (αβ + βγ + γα) = ?
–1
1
–5
30
If α, β, γ are the zeros of the polynomial 2x3 + x2 – 13x + 6 then αβγ = ?
–3
3
`(-1)/2`
`(-13)/2`
If α, β, γ be the zeros of the polynomial p(x) such that (α + β + γ) = 3, (αβ + βγ + γα) = –10 and αβγ = –24 then p(x) = ?
x3 + 3x2 – 10x + 24
x3 + 3x2 + 10x – 24
x3 – 3x2 – 10x + 24
None of these
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are 0 then the third zero is ______.
`(-b)/a`
`b/a`
`c/a`
`(-d)/a`
If one of the zeros of the cubic polynomial ax3 + bx2 + cx + d is 0 then the product of the other two zeros is ______.
`(-c)/a`
`c/a`
0
`(-b)/a`
If one of the zeros of the cubic polynomial x3 + ax2 + bx + c is –1 then the product of the other two zeros is ______.
a – b – 1
b – a – 1
1 – a + b
1 + a – b
If α, β be the zeros of the polynomial 2x2 + 5x + k such that `α^2 + β^2 + αβ = 21/4` then k = ?
3
–3
–2
2
On dividing a polynomial p(x) by a nonzero polynomial q(x), let g(x) be the quotient and r(x) be the remainder then p(x) = q(x) · g(x) + r(x), where
r(x) = 0 always
deg r(x) < deg g(x) always
either r(x) = 0 or deg r(x) < deg g(x)
r(x) = g(x)
Which of the following is a true statement?
x2 + 5x – 3 is a linear polynomial.
x2 + 4x – 1 is a binomial.
x + 1 is a monomial.
5x3 is a monomial.
R.S. Aggarwal solutions for Mathematics [English] Class 10 2 Polynomials TEST YOURSELF [Pages 75 - 76]
MCQ
Zeros of p(x) = x2 – 2x – 3 are ______.
1, –3
3, –1
–3, –1
1, 3
If α, β, γ are the zeros of the polynomial x3 – 6x2 – x + 30 then the value of (αβ + βy + γα) is ______.
–1
1
–5
30
If α, β are the zeros of kx2 – 2x + 3k such that α + β = αβ then k = ?
`1/3`
`(-1)/3`
`2/3`
`(-2)/3`
It is given that the difference between the zeros of 4x2 – 8kx + 9 is 4 and k > 0. Then, k = ?
`1/2`
`3/2`
`5/2`
`7/2`
Short-Answer Questions
Find the zeros of the polynomial x2 + 2x – 195.
If one zero of the polynomial (a2 + 9)x2 + 13x + 6a is the reciprocal of the other, find the value of a.
Find a quadratic polynomial whose zeros are 2 and –5.
If the zeros of the polynomial x3 – 3x2 + x + 1 are (a – b), a and (a + b), find the values of a and b.
Verify that 2 is a zero of the polynomial x3 + 4x2 – 3x – 18.
Find the quadratic polynomial, the sum of whose zeros is –5 and their product is 6.
Find a cubic polynomial whose zeros are 3, 5 and –2.
Using remainder theorem, find the remainder when p(x) = x3 + 3x2 – 5x + 4 is divided by (x – 2).
Show that (x + 2) is a factor of f(x) = x3 + 4x2 + x – 6.
If α, β, γ are the zeros of the polynomial p(x) = 6x3 + 3x2 – 5x + 1, find the value of `(1/α + 1/β + 1/γ)`.
If α, β are the zeros of the polynomial f(x) = x2 – 5x + k such that α – β = 1, find the value of k.
Show that the polynomial f(x) = x4 + 4x2 + 6 has no zero.
Long-Answer Questions
If one zero of the polynomial p(x) = x3 – 6x2 + 11x – 6 is 3, find the other two zeros.
If two zeros of the polynomial p(x) = 2x4 – 3x3 – 3x2 + 6x – 2 are `sqrt(2)` and `-sqrt(2)`, find its other two zeros.
Find the quotient when p(x) = 3x4 + 5x3 – 7x2 + 2x + 2 is divided by (x2 + 3x + 1).
Use remainder theorem to find the value of k, it being given that when x3 + 2x2 + kx + 3 is divided by (x – 3) then the remainder is 21.
Solutions for 2: Polynomials
R.S. Aggarwal solutions for Mathematics [English] Class 10 chapter 2 - Polynomials
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