### Polynomials Class 10th Mathematics Gujarat Board Solution

##### Question 1.State whether the following statements are true or false:(1) is a zero of the linear polynomial p(x) = 5x + 7.(2) p(x) = x2 + 2x + 1 has two distinct zeros.(3) The cubic polynomial p(x) = x3 + x2 — x — 1 has two distinct zeros.(4) The graph of the cubic polynomial p(x) = x3 meets the X—axis at only one point.(5) Any quadratic polynomial p(x) has at least one zero, x RAnswer:(1) p(x) = 5x + 7∴ p = 5 + 7 = 14⇒ p ≠0⇒ is not a zero of p(x).So the statement is False.(2) Given, p(x) = x2 + 2x + 1To find the zeros of p(x), let p(x) = 0∴ x2 + 2x + 1 = 0∴ (x + 1)2 = 0 Using the identity: (a+ b) 2 = (a2 + 2ab + b2)∴ x + 1 = 0 or x = –1∴ x = –1 or x = –1Here, both the zeros are equal, i.e. –1, and hence not distinct.So the statement is False.(3) Given, p(x) = x3 + x2 — x — 1To find the zeros of p(x), let p(x) = 0∴ x3 + x2 — x — 1 = 0∴ x2 (x + 1) – 1 (x + 1) = 0∴ (x2 – 1)(x + 1) = 0∴ (x – 1) (x + 1)(x + 1) = 0 Using the identity: (a2 – b2) = (a – b) (a + b)∴ x – 1 = 0 or x + 1 = 0 or x + 1 = 0∴ x = 1 or x = –1 or x = –1∴ Two distinct zeros of p(x) are 1 and –1.Hence, p(x) has at the most two distinct zeros.So the statement is True.(4) Given, p(x) = x3To find the zeros of p(x), let p(x) = 0∴x3 = 0⇒ x = 0.⇒ The graph of p(x) = x3 meets the X-axis at only one point i.e. (0, 0).So the statement is True.(5) If the graph of the quadratic polynomial p(x) does not intersect the x-axis at any point, then the quadratic polynomial does not have any real zero.So the statement is False.Question 2.Find the zeros and number of zeros of p(x) = x2 + 9x + 18. Show them on a graph.Answer:Here, p(x) = x2 + 9x + 18To find the zeros of p(x), let p(x) = 0.∴ x2 + 9x + 18 = 0∴ x2 + 6x + 3x + 18 = 0∴ x(x + 6) + 3(x + 6) = 0∴ (x + 6)(x + 3) = 0∴ x = –6 or x = –3Thus, the number of zeros of p(x) = 2.To draw the graph of this polynomial, consider the following values of x and corresponding values of p(x): On plotting the above points on a graph paper we obtain the following shape of the graph: Question 3.Find the zeros, the sum and the product of zeros of p(x) = 4x2 + 12x + 5.Answer:Given, p(x) = 4x2 + 12x + 5= 4x2 + 10x + 2x + 5= 2x(2x + 5) + 1(2x + 5)= (2x + 5)(2x + 1)To find the zeros of p(x), put p(x) = 0∴ (2x + 5)(2x + 1) = 0∴ x = – or x = – The zeros of p(x) are – and – .Now, sum of zeros = = = –3and, product of zeros = – × – = Question 4.—4 and 9 are the sum and product of the zeros respectively of a quadratic polynomial. Find the quadratic polynomial.Answer:Let α and β be the zeros of the polynomial p(x) = ax2 + bx + c.Given, α + β = –4 ; αβ = 9A quadratic polynomial with respect to its roots is given by:p(x) = x2 - (α + β) x + αβ = 0Putting the values in this equation we get,p(x) = x2 - (- 4) x + 9p(x) = x2 + 4x + 9Question 5.Find q(x) and r(x), for the quadratic polynomial p(x) = 11x — 21 + 2x2 when divided by 1 + 2xAnswer:Here, dividend polynomial = p(x) = 11x — 21 + 2x2 = 2x2 + 11x — 21and divisor polynomial = s(x) = 1 + 2x = 2x + 1 The quotient polynomial q(x) = x + 5 and the reminder polynomial r(x) = –26.Question 6.Divide 2x3 + 3x2 — 11x — 6 by x2 + x — 6Answer:Here, dividend polynomial = p(x) = 2x3 + 3x2 — 11x — 6and divisor polynomial = s(x) = x2 + x — 6 Thus, the quotient polynomial q(x) = 2x + 1 and the reminder polynomial r(x) = 0.Question 7.4 is a zero of the cubic polynomial p(x) = x3 — 3x2 — 6x + 8. Find the remaining zeros of p(x).Answer:Given, 4 is a zero of polynomial p(x).So (x – 4) is the factor of p(x).Here, dividend polynomial = p(x) = x3 — 3x2 — 6x + 8and divisor polynomial = s(x) = x – 4.Coefficients of x3, x2, x and x° are 1, –3, –6 and 8 respectively.Taking x – 4 = 0 we get x = 4 ∴ p(x) = x3 — 3x2 — 6x + 8= (x – 4) (x2 + x – 2)= (x – 4) (x2 – x + 2x – 2)= (x – 4) (x(x – 1) + 2(x –1))= (x – 4)(x + 2)(x – 1)To find the remaining zeros, let p(x) = 0i.e. (x – 4)(x + 2)(x – 1) = 0∴ The remaining zeros of p(x) are –2 and 1.Question 8.The product of two polynomials is 3x4 + 5x3 — 21x2 — 53x — 30. If one of them is x2 — x — 6, find the other polynomial.Answer:Here p(x) = the dividend polynomial = 3x4 + 5x3 — 21x2 — 53x — 30s(x) = the divisor polynomial = x2 — x — 6Now, ∴ The quotient polynomial q(x) = 3x2 + 8x + 5 and the remainder polynomial r(x) = 0.∴ The other polynomial is 3x2 + 8x + 5.Question 9.2 + √3 and 2 — √3 are the zeros of p(x) = x4 — 6x3 — 26x2 + 138x — 35. Find the remaining zeros of p(x).Answer:Given, 2 + √3 and 2–√3 are zeros of p(x).⇒ and are the factors of p(x).Thus = (x2 – 4x + 4 –3) = (x2 – 4x + 1) is also a factor of p(x).To find the remaining zeros, we find the remaining factors using the division process.Here, dividend polynomial = p(x) = 2x4 + 7x3 — 8x2 — 14x + 8and divisor polynomial = s(x) = x2 – 2 ⇒ p(x) = 2x4 + 7x3 — 8x2 — 14x + 8 = (x2 – 4x + 1)(x2 – 2x — 35)On factorising x2 – 2x — 35, we getx2 – 2x — 35 = x2 –7x + 5x— 35= x (x – 7) + 5 (x – 7)= (x – 7) (x + 5)Hence the other two zeros of p(x) are 7 and –5.Question 10.The linear polynomial p(x) = 7x — 3 has the zero _____A. B. C. – D. – Answer:Here, p(x) = 7x — 3To find the zeros of p(x), consider p(x) = 0∴ 7x — 3 = 0∴ x = Hence, the zero of the given polynomial is .The correct option is B.Question 11.The cubic polynomial p(x) = x3 — x has _____ zeros.A. 0B. 1C. 2D. 3Answer:Here, p(x) = x3 — xTo find the zeros of p(x), consider p(x) = 0∴x3 — x = 0∴ x (x2 – 1) = 0∴ x(x – 1)(x + 1) = 0 Using the identity: (a2 – b2) = (a – b) (a + b)∴ x = 0, x = 1, x = –1∴ The cubic polynomial has 3 zeros.The correct option is D.Question 12.The graph of p(x) = 3x — 2 — x2 intersects the X-axis in _____ points.A. 0B. 1C. 2D. 3Answer:The zeros of p(x) are the intersections points of the equation, p(x) = 3x – 2 – x2 with the x-axis.∴ The number of distinct zeros of p(x) gives the number of distinct intersection points on the X-axis.To find the zeros of p(x), consider p(x) = 0∴3x – 2 – x2 = 0∴ x2 – 3x+ 2 = 0∴ x2 – x – 2x+ 2 = 0∴ x(x – 1) – 2(x – 1) = 0∴ (x – 1)(x – 2) = 0∴ x = 1 or x = 2 are the zeros of p(x).⇒The graph of p(x) intersects X-axis at two points.The correct option is C.Question 13.The sum of the zeros of 3x2 + 5x — 2 is _____A. B. – C. D. – Answer:Given, p(x) = 3x2 + 5x — 2Here a = 3, b = 5, c = –2Sum of zeros, α + β = – = – The correct option is D.Question 14.The graph of p(x) = 3x + 5 represents _____A. a straight lineB. parabola open upwardsC. parabola open downwardsD. a rayAnswer:Given, p(x) = 3x+ 5Since p(x) is a linear polynomial, so its graph is a straight line.The correct option is A.Question 15.A quadratic polynomial has no zero. Its graph _____A. touches X-axis at any pointB. intersects X-axis at two distinct pointsC. does not intersect X-axis at two distinct pointsD. is in any one half plane of X-axisAnswer:The number of intersection points of a quadratic polynomial at X-axis is the number of its real zeros.Given, the quadratic polynomial has no zero.⇒ Its graph does not intersect X-axis or its graph lies in any one half plane of X-axis.The correct option is D.Question 16.For the graph in figure 2.11 y = p(x) has _____ zeros. A. 1B. 2C. 3D. 4Answer:The graph intersects the X-axis at four distinct points.⇒ p(x) has four zeros.The correct option is D.Question 17.The product of the zeros of x2 — 4x + 3 is _____A. 1B. 3C. 4D. —4Answer:Given, p(x) = x2 — 4x + 3Here a = 1, b = –4, c = 3Product of zeros, αβ = = – = 3The correct option is B.Question 18.a = 3, b = 5, c = 7, d = 11 in the standard notation gives the cubic polynomial _____A. 3x3 + 5x2 — 7x — 11B. 3x3 — 5x2 + 7x — 11C. 3x3 + 5x2 — 7x + 11D. 3x3 + 5x2 + 7x + 11Answer:The standard form of a cubic polynomial is p(x) = ax3 + bx2 + cx + d∴ For a = 3, b = 5, c = 7 and d = 11,p(x) = 3x3 + 5x2 + 7x + 11.The correct option is D.

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