# Worksheets For Class 10 Mathematics Polynomials

Polynomials Class 10 Worksheet have been designed as per the latest pattern for CBSE, NCERT and KVS for Grade 10. Students are always suggested to solve printable worksheets for Mathematics Polynomials Grade 10 as they can be really helpful to clear their concepts and improve problem solving skills. We at worksheetsbag.com have provided here free PDF worksheets for students in standard 10 so that you can easily take print of these test sheets and use them daily for practice. All worksheets are easy to download and have been designed by teachers of Class 10 for benefit of students and is available for free download.

## Mathematics Polynomials Worksheets for Class 10

We have provided chapter-wise worksheets for class 10 Mathematics Polynomials which the students can download in Pdf format for free. This is the best collection of Mathematics Polynomials standard 10th worksheets with important questions and answers for each grade 10th Mathematics Polynomials chapter so that the students are able to properly practice and gain more marks in Class 10 Mathematics Polynomials class tests and exams.

PRACTICE EXERCISE

Question. Draw the graph of each of the following polynomials and if possible, read the zero(s) from the graph :
(a) x2 – 2x + 9 (b) –2x2 + 4x
(c) x2 + 2x – 3 (d) x2 – 8x + 16
(e) x3 (f) x3 – 2x2
Solution. (a) No zero (b) 0 and 2 (c) –3 and 1 (d) 4 (e) 0 (f) 0 and 2

Question. Draw the graph of the polynomial x2 – 3x – 10. Read off the zeroes of the polynomial from the graph. Also show the axis of symmetry on it.
Solution. zeroes 5 and –2 ; Axis of symmetry; x = 3/2

Question. Find the zeroes of each of the following quadratic polynomial. Also, in each case, verify the relationship between the zeroes and its coefficients :
(a) x2 + 8x + 12 (b) x2 + 3x – 4
(c) x2 – 7x + 10 (d) y2 – 4
(e) 4u2 – 1 (f) 3t2 – 5
Solution. (a) –2 and –6 (b) 1 and –4 (c) 2 and 5 (d) ± 2 (e) ± 1/2 (f) ±√5/3

Question. On dividing x3 – 3x2 + x + 2 by a polynomial g(x), the quotient and remainder are (x – 2) and (–2 x + 4) respectively. Find g(x).
Solution. x2 – x + 1

Question. Find a quadratic polynomial whose sum and product of the zeroes are 3 and –5 respectively.
Solution. x2 – 3x – 5

Question. If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a, a + b, find a and b.
Solution. a = 1, b = ± √2

Question. Divide p(x) = 9x4 – 4x2 + 4 by q(x) = 3x2 + x – 1. Also, find quotient and remainder.
Solution. quotient = 3x2 – x, remainder = – x + 4

Question. Find zeroes of f (x)= x2 + 5x – 14. Also, verify the relationship between zeroes and its coefficients.
Solution. 2 and – 7

Question. Find all the zeroes of the polynomial f(x) = 2x4 – 3x3 – 3x2 + 6x – 2, if its two zeroes are – √2 and √2 .
Solution. – √2 , √2 , 1 , 1/2

Question. The graph of y = p(x) are given below. find the number of zeroes of p(x).

Solution. (a) 4 (b) 2

Question. Find a quadratic polynomial each with the given numbers as the sum and the product of its zeroes respectively :
(a) 3 and 4 (b) –2 and 3/ 2
(c) – 3/2 and 0 (d) – √2 and √3
Solution. (a) x2 – 3x + 4 (b) x2 + 2x + 3/2 (c) x2 + 3/2 x  (d) x2 + √2x + √3

Question. If the polynomial x4 – 6x3 + 16x2 – 26x + 10 – a is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a. Find k and a.
Solution. k = 5, a = – 5

Question. Obtain all the zeroes of the polynomial 2x4 – 2x3 – 7x2 + 3x + 6 ; if two of its zeroes are ± √3/2 .
Solution. 2, –1

Question. Find a cubic polynomial with the sum, sum of product of its zeroes taken two at a time and the product of its zeroes respectively as given below :
(a) –4, 7 and 0 (b) 5, –2 and – 24
(c) – 2, – 8/3 and 0
Solution. (a) x3 + 4x2 – 7 (b) x3 – 5x2 – 2x + 24 (c) 3x3 – 6x + 8

Question. Without drawing actual graph, find the zeroes of the polynomials if any. Give reason.
(a) x2 – 2x– 8 (b) –x2 – 2x + 3
(c) x2 + x + 1 (d) x2 – 1
(e) x2 + 4x + 4 (f) –4x2+ 4x – 1
Solution. (a) –2 and 4 (b) –3 and 1 (c) no zero (d) 1 and –1 (e) –2 and –2 (f) 1/2 and 1/2

Question. The graphs of y = p(x) are given below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

Solution. (a) 1 (b) 1 (c) 2 (d) 2 (e) 1 (f) 1

Question. Write a cubic polynomial with zeroes : – 2, 1, 4 and hence find :
(a) sum of its zeroes (b) sum of its zeroes taking any two at a time
(c) product of three zeroes
Solution. x3 – 3x2 – 6x + 8; (a) 3 (b) –6 (c) –8

Question. The graph of y = p(x) are given below, for some polynomial p(x). Find the zeroes of the corresponding polynomial.

Solution. (a) –2, 4 (b) –2, 3 (c) –2, 0, 2 (d) –1.5, 2, 4

Question. Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm :
(a) x + 8; x3 + 15 x2 + 56x (b) x – 2; x4 – 3x3 + 3x – 9
(c) x2 – 2; x3 – 3x2 + 5x – 3 (d) – 5y2 – 4y + 2; 15y4 + 2y3 – 39y2 – 16y + 10
(e) 2x2 + 3x + 4 ; 2x4 + 7x3 + 10x2 + 14x + 8 (f) x3 – 3x + 1; x5 – 4x3 + x2 + 3x + 1
Solution. (a) yes (b) yes (c) no (d) yes (e) no (f) no

Question. What must be added to the polynomial p(x) = x4 + 2x3 – 2x2 + x – 1 so that the resulting polynomial is exactly divisible by x2 + 2x – 3?
Solution. x – 2

Question. Find the value of a and b so that 1, –2 are the zeroes of the polynomial x3 + 10x2 + ax + b.
Solution. a = 7, b = – 18

Question. On dividing x3 – 3x2 + 5x – 3 by a polynomial g(x), the quotient and remainder are x – 3 and 7x – 9 respectively. Find g(x).
Solution. x2 – 2

Question. Obtain all the zeroes of the polynomial f(x) = x4 – 3x3 – x2 + 9x – 6; if two of its zeroes are √3 and – √3 .
Solution. – √3, √3, 1, 2

Question. On dividing x4 – 5x + 6 by a polynomial g(x), the quotient and remainder are –x2 – 2 and –5x + 10 respectively. Find g(x).
Solution. – x2 + 2

Question. Apply the division algorithm to find the quotient and the remainder on division of p(x) by g(x) as given below :
(a) p(x) = –5x2 + 14x3 + 9x – 1, g(x) = –1 + 2x (b) p(x) = 6x3 + 11x2 – 39x – 65, g(x) = x2 + x – 1
(c) p(x) = x4 – 5x + 6, g(x) = 2 – x2 (d) p(x)= 3x3 + x2+ 2x + 5, g(x) = 1 + 2x + x2
(e) p(x) = 4x3 – 27x + 4x2 + 16, g(x) = 2x – 3 (f) p(x) = 6x3 + 19x2 + 18x – 5, g(x) = 3x + 5
(g) p(x) = –2x4 – 12x3 – 22x2 – 17x + 4, g(x) = x2 – 3x + 4
(h) p(x) = 6x5 + 4x4 – 3x3 + x + 1, g(x) = 3x2 – x + 1
Solution. (a) quotient = 7x2 + x + 5, remainder = 4
(b) quotient = 6x + 5, remainder = – 38x – 60
(c) quotient = – x2 – 2, remainder = – 5x + 10
(d) quotient = 3x – 5, remainder = 9x + 10
(e) quotient = 2x2 + 5x – 6, remainder = – 2
(f) quotient = 2x2 + 3x + 1, remainder = – 10
(g) quotient = – 2x2 – 9x – 49, remainder = 130 x + 200
(h) quotient = 2x3 + 2x2 – x – 1, remainder = x + 2

Question. What must be subtracted from 8x4 + 14x3 – 2x2 + 7x – 8 so that the resulting polynomial is exactly divisible by 4x2 + 3x – 2?
Solution. 14 x – 10

Question. If the polynomial f (x) = x 4 – 6x3 +16x2 – 25x +10 is divided by another polynomial x2 – 2x + k, the remainder comes out to be x + a. Find k and a.
Sol. By di vi sion al go rithm, we have
Dividend = Divisor ´ Quotient + Remainder
⇒ Dividend – Remainder = Divisor x Quotient
⇒ Dividend – Remainder is always divisible by the divisor.
When f (x) = x 4 – 6x3 +16x2 – 25x +10 is divided by x 2 – 2x + k the remainder comes out to be x + a .
∴ f (x) – (x + a) = x4 – 6x3 +16x2 – 25x +10 -(x + a)
= x 4 – 6x3 +16x2 – 25x +10 – x – a
= x 4 – 6x3 +16x2 – 26x +10 – a
is exactly divisible byx2 – 2x + k.
Let us now divide x 4 – 6x3 +16x2 – 26x +10 – a byx2 – 2x + k.

For f (x) – (x + a) = x4 – 6x3 +16x 2 – 26x +10 – a to be exactly divisible by x2 – 2x + k, we must have
(-10 + 2k) x + (10 – a – 8k + k2) = 0 for all x
⇒ -10 + 2k = 0 and 10 – a – 8k + k2 = 0
⇒ k = 5 and 10 – a – 40 + 25 = 0
⇒ k = 5 and a = – 5.

(i) Zeroes are – √2 – 1/2,          (ii) g(x) = -4x2 -3x +6

Question. What must be subtracted from p(x) = 8x 4 +14x 3 – 2x 2 + 7x – 8 so that the resulting polynomial is exactly divisible by g(x) = 4x 2 + 3x – 2?
Sol. Let y be sub tracted from polynomial p (x)
∴ p(x) = 8x4 +14x3 – 2x2 +7x – 8 – y

Q Remainder should be 0.
∴ 14x -10 – y = 0
or 14x -10 = y or    y =14x -10
∴ (14x -10) should be subtracted from p(x) so that it will be exactly divisible by g(x).

Question. (i) If the remainder on division of x 3 -2x2 + kx +5 by x -2 is 11, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x3 -2x2 + kx -6.
(ii) Given that the zeroes of the cubic polynomial x 3 -6x2 +3x +10 are of the form a, a + b, a +2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

(i) k = 3, Quotient = x2 +3, 2 is the zero of x 3 -2x2 +3x-6
(ii) When a = 5, b = -3 and when a = -1, b = 3, zeroes are –1, 2, 5

Question State whether the following statements are true or false. Justify your answer.
(i) If the zeroes of a quadratic polynomial ax 2 + bx + c are both positive, then a, b and c all have the same sign.
(ii) The quotient and remainder on division of 2x2 +3x + 4by x 3 +1 are 0 and 2x2 +3x + 4 respectively.

(i) False    (ii) True

Question. Divide the polynomial f(x) = x4 – 3x2 + 4x + 5 by the polynomial 1 – x + x2 and verify the division algorithm.
Solution. On writing the dividend and the divisor in the standard form, we get :
Dividend p(x) = x4 – 3x2 + 4x + 5 and divisor g(x) = x2 – x + 1.
Now,

Clearly, quotient = x2 + x – 3 and remainder = 8.

Question. Find the zeroes of the quadratic polynomial x2 – 2x – 8, and verify the relationship between the zeroes and their coefficients.
Solution. Let f(x) = x2 – 2x – 8
The zeroes of f(x) are given by f(x) = 0.
⇒ x2 – 2x – 8 = 0
⇒ x2 – 4x + 2x – 8 = 0
⇒ x (x – 4) + 2 (x – 4) = 0
⇒ (x + 2) (x – 4) = 0
⇒ x = – 2 or x = 4 Ans.

Question. Obtain all the zeroes of the polynomial f (x) = 3x4 + 6x3 – 2x2 – 10x – 5, if two of its zeroes are

Solution.

Question. Find a quadratic polynomial whose sum and product of the zeroes is 1/4 and –1 respectively .
Solution. We know that a quadratic polynomial when the sum and product of its zeroes are given by f(x) = k {x2 – (sum of the zeroes) x + Product of the zeroes}, where k is a constant.
∴ Required quadratic polynomial f(x) is given by f (x) = k (x2 – 1/4 , x-1)
OR
Let the required quadratic polynomial be ax2 + bx + c.
Since, sum of zeroes = -b/a ⇒ 1/4 = -b/a
and, product of zeroes = c/a ⇒ -1 = c/a ⇒ -4/4 = c/a
If a = 4; then b = 1 and c = – 4.
∴ Required quadratic polynomial = ax2 + bx + c
= 4x2 – x – 4

Question. Draw the graph of the polynomial f(x) = x2 – 2x – 3. Obtain the vertex of this parabola. Also, read the zeroes of the polynomial, if possible from the graph.
Solution.
Let y = f(x) = x2 – 2x – 3.
The following table gives the values of y for various values of x.

After plotting the points in the table on a graph paper, draw a free-hand continuous curve (parabola) through all the points plotted.

Vertex of Parabola : On comparing the polynomial x2 – 2x – 3 with ax2 + bx + c, we get a = 1,
b = –2, c = – 3. The vertex of the parabola has the co-ordinates (-b/2a , – D/4a) , where D = b2 – 4ac.
Now, D = b2 – 4ac = (–2)2 – 4 (1) (–3)= 4 + 12 = 16
∴  Co-ordinate of vertex is [-(-2)/2(1) ,-16/4(1)] = (1,4)
Zeroes of f(x) = x2 – 2x – 3
Since the parabola f(x) = x2 – 2x – 3 cuts the x-axis at (–1, 0) and (3, 0).
∴ zeroes of f(x) are –1 and 3.

## Mathematics Polynomials Worksheets for Class 10 as per CBSE NCERT pattern

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