2. Polynomials
рдЗрд╕ рдкреЗрдЬ рдкрд░ рдЖрдкрдХреЛ 2. Polynomials рдХреЗ рд╕рднреА рдорд╣рддреНрд╡рдкреВрд░реНрдг рдмрд╣реБрд╡рд┐рдХрд▓реНрдкреАрдп рдкреНрд░рд╢реНрди (MCQ), рдкрд░реАрдХреНрд╖рд╛ рдЙрдкрдпреЛрдЧреА рдиреЛрдЯреНрд╕ рдФрд░ рдСрдирд▓рд╛рдЗрди рдХреНрд╡рд┐рдЬрд╝ рдорд┐рд▓реЗрдВрдЧреЗред рдЕрдкрдиреЗ рдЬреНрдЮрд╛рди рдХрд╛ рдкрд░реАрдХреНрд╖рдг рдХрд░рдиреЗ рдФрд░ рдмреЛрд░реНрдб/рдкреНрд░рддрд┐рдпреЛрдЧреА рдкрд░реАрдХреНрд╖рд╛ рдореЗрдВ рд╢рдд-рдкреНрд░рддрд┐рд╢рдд рдЕрдВрдХ рдкреНрд░рд╛рдкреНрдд рдХрд░рдиреЗ рдХреЗ рд▓рд┐рдП рдиреАрдЪреЗ рджрд┐рдП рдЧрдП рдореЙрдХ рдЯреЗрд╕реНрдЯ рдореЗрдВ рдЕрд╡рд╢реНрдп рднрд╛рдЧ рд▓реЗрдВред
ЁЯСЙ рдЕрдзреНрдпрд╛рдп рдХрд╛ рд╕рд╛рд░рд╛рдВрд╢ (Summary)
Dear students, welcome to Chapter 2 of Class 10 Mathematics, Polynomials. You have already studied polynomials in one variable and their degrees in Class 9. In this chapter, we will take a deeper dive into the world of polynomials. We begin by understanding the geometrical meaning of the zeroes of a polynomial. The zeroes of a polynomial p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis. This visual representation provides a profound understanding of what solving a polynomial equation actually means. We will explore linear polynomials, which represent straight lines and have exactly one zero; quadratic polynomials, which form parabolic curves and have at most two zeroes; and cubic polynomials, which can intersect the x-axis at up to three points.
A significant portion of this chapter is dedicated to uncovering the elegant mathematical relationship between the zeroes and the coefficients of a polynomial. For a quadratic polynomial ax^2 + bx + c, the sum of its zeroes is strictly equal to -b/a, and the product of its zeroes is equal to c/a. This powerful algebraic property allows us to seamlessly construct polynomials if we know their roots, and vice versa. We will also extend this fundamental concept to cubic polynomials, exploring the sum of zeroes, the sum of the product of zeroes taken two at a time, and the product of all three zeroes. These relationships are foundational building blocks for higher algebra.
Furthermore, we will learn about the Division Algorithm for polynomials, which states that if p(x) and g(x) are any two polynomials with g(x) тЙа 0, then we can find polynomials q(x) and r(x) such that p(x) = g(x)q(x) + r(x), where r(x) = 0 or the degree of r(x) < degree of g(x). This algorithm is extremely useful in finding all the zeroes of a higher-degree polynomial if some of its zeroes are already given to us. Mastering polynomials is absolutely crucial as they are extensively used in modeling various real-world situations, from predicting the continuous trajectory of a thrown object to calculating the structural area and volume of complex designs. Keep practicing the diverse algebraic manipulations, as they are a staple in board exams.
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рдорд╣рддреНрд╡рдкреВрд░реНрдг рдкреНрд░рд╢реНрди (Important Questions)
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*рдиреЛрдЯ: рдпрд╣ рд╕реЗрдХреНрд╢рди рд░рд┐рд╡реАрдЬрди рдХреЗ рд▓рд┐рдП рд╣реИред рдЕрдкрдирд╛ рдЬреНрдЮрд╛рди рдкрд░рдЦрдиреЗ рдХреЗ рд▓рд┐рдП рдКрдкрд░ рджрд┐рдП рдЧрдП рдХреНрд╡рд┐рдЬрд╝ рдореЗрдВ рднрд╛рдЧ рд▓реЗрдВред
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What is the degree of a non-zero constant polynomial?
A) 0 | B) 1 | C) 2 | D) Not defined -
A polynomial of degree 2 is called:
A) Linear polynomial | B) Quadratic polynomial | C) Cubic polynomial | D) Biquadratic polynomial -
What is the maximum number of zeroes a quadratic polynomial can have?
A) 1 | B) 2 | C) 3 | D) Infinitely many -
Which of the following algebraic expressions is a polynomial?
A) x + 1/x | B) sqrt(x) + 3 | C) x^2 - 3x + 2 | D) 1/(x-1) -
The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the:
A) x-axis | B) y-axis | C) Origin | D) Both axes -
Find the value of the polynomial p(x) = x^2 - 5x + 6 at x = 2.
A) 0 | B) 1 | C) 2 | D) 6 -
If x = 1 is a zero of the polynomial p(x) = x^2 - kx + 4, then the value of k is:
A) 3 | B) 4 | C) 5 | D) -5 -
What is the degree of the polynomial 5x^3 - 4x^2 + x - sqrt(2)?
A) 1 | B) 2 | C) 3 | D) 4 -
A linear polynomial has:
A) No zero | B) Exactly one zero | C) Two zeroes | D) Three zeroes -
The graph of a linear polynomial is a:
A) Parabola | B) Straight line | C) Circle | D) Ellipse -
If alpha and beta are the zeroes of the quadratic polynomial ax^2 + bx + c, then what is the value of alpha + beta?
A) c/a | B) -c/a | C) b/a | D) -b/a -
Find the sum of the zeroes of the quadratic polynomial x^2 - 7x + 10.
A) 7 | B) -7 | C) 10 | D) -10 -
Find the product of the zeroes of the quadratic polynomial 3x^2 - 5x - 2.
A) 5/3 | B) -5/3 | C) 2/3 | D) -2/3 -
A quadratic polynomial whose sum and product of zeroes are -3 and 2 respectively is:
A) x^2 - 3x + 2 | B) x^2 + 3x + 2 | C) x^2 + 2x - 3 | D) x^2 - 2x - 3 -
Find the zeroes of the polynomial x^2 - 3.
A) 3 and -3 | B) sqrt(3) and -sqrt(3) | C) 3 and 1 | D) sqrt(3) and 1 -
If alpha and beta are the zeroes of f(x) = x^2 + x + 1, then the value of 1/alpha + 1/beta is:
A) 1 | B) -1 | C) 0 | D) 2 -
If one zero of the quadratic polynomial x^2 + 3x + k is 2, then the value of k is:
A) 10 | B) -10 | C) 5 | D) -5 -
The graph of a quadratic polynomial ax^2 + bx + c (where a > 0) is an upward opening:
A) Straight line | B) Circle | C) Parabola | D) Hyperbola -
If one zero of the polynomial (k-1)x^2 + kx + 1 is -3, then the value of k is:
A) 4/3 | B) -4/3 | C) 2/3 | D) -2/3 -
If the product of the zeroes of the polynomial ax^2 - 6x - 6 is 4, find the value of a.
A) -3/2 | B) 3/2 | C) -2/3 | D) 2/3 -
If alpha, beta, gamma are the zeroes of the cubic polynomial ax^3 + bx^2 + cx + d, then the product alpha*beta*gamma is equal to:
A) -b/a | B) c/a | C) -d/a | D) d/a -
According to the division algorithm, p(x) = g(x)q(x) + r(x). What is the mathematical condition for the remainder r(x)?
A) r(x) = 0 only | B) deg r(x) > deg g(x) | C) deg r(x) < deg g(x) or r(x) = 0 | D) deg r(x) = deg g(x) -
If the zeroes of the quadratic polynomial ax^2 + bx + c (c тЙа 0) are equal in magnitude but of the same sign, then:
A) c and a have opposite signs | B) c and b have opposite signs | C) c and a have the same sign | D) c and b have the same sign -
If the zeroes of the polynomial x^3 - 3x^2 + x + 1 are a - b, a, and a + b, find the value of a.
A) 0 | B) 1 | C) -1 | D) 2 -
If alpha and beta are the zeroes of 2x^2 + 5x + 1, then the value of alpha^2 + beta^2 + alpha*beta is:
A) 21/4 | B) 23/4 | C) 25/4 | D) 27/4 -
Find a quadratic polynomial whose zeroes are exactly the reciprocals of the zeroes of the polynomial ax^2 + bx + c.
A) cx^2 + bx + a | B) cx^2 - bx + a | C) ax^2 - bx + c | D) bx^2 + cx + a -
If alpha and beta are zeroes of the polynomial x^2 - 5x + k such that alpha - beta = 1, find the value of k.
A) 4 | B) 5 | C) 6 | D) 7 -
What must be strictly subtracted from x^2 - 6x + 8 to make it exactly divisible by x - 2?
A) 0 | B) 1 | C) 2 | D) -2 -
A given polynomial of degree n always has:
A) Exactly n zeroes | B) At least n zeroes | C) At most n zeroes | D) Infinite zeroes -
If the sum of the squares of the zeroes of the quadratic polynomial f(x) = x^2 - 8x + k is equal to 40, find the value of k.
A) 10 | B) 12 | C) 14 | D) 16
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