Tenth Maths Polynomials Polynomials are algebraic expressions constructed using constants and variables. Coefficients operate on variables, which can be raised to various powers of non-negative integer exponents. 1 1. Solomon said that is a polynomial of degree two as the exponent of x 2 the variable x is 2 a non - negative integer. Do you agree? Hint: The exponent of ( 1 x ) is 2, but not of 'x'. 1 = x -2 x 2 The exponent of x is 2 which is a negative integer. 2. Radhika said that every quadratic polynomial has exactly two zeroes and Nikhat argued that a quadratic polynomial has at the most two zeroes. What do you understand from their statements? Explain (The zeroes under consideration are real numbers). Hint: Observe the following three quadratic polynomial. (i) x 2 3x + 2 (ii) x 2 + 2x + 1 (iii) x 2 + 4x + 5 How many zeroes each of the above polynomial has? Write your conclusion based on the number of zeroes of the above polynomials. 3. If two zeroes of the polynomial x 4 2x 3 8x 2 6x + 1 are 2 ± 5. Find the remaining zeroes. Hint: The given two zeroes of the polynomial are 2 + 5 and 2 5. Frame a R-11-1-15 quadratic polynomial with these zeroes. Divide the given polynomial with this quadratic polynomial. Find the zeroes of the quadratic polynomial that is obtained as quotient in the division.
4. Mukkram observed the following graph. And said that there is only one zero of the polynomial p(x) as the graph y = p(x) intersects the X - axis at only one point. Hence p(x) is a linear polynomial. Do you agree? Why / Why not? Hint: Graph of y = p(x) where p(x) is a linear polynomial is a straight line. Observe whether the given graph is a straight line. Observe the relation between the x-coordinate and y-coordinate in the given ordered pairs ( 2, 8), ( 1, 1) (0, 0), (1,1) and (2, 8). Try to write the polynomial from the relation you observed between the x-coordinate and y-coordinate, in the ordered pairs. Write the degree of the polynomial stating your reasons. 5. If α and β are the zeroes of a polynomial ax 2 + bx + c then find the quadratic polynomial whose zeroes are and. Hint: Write α + β and αβ in terms of 'a' and 'b' Find the sum of the zeroes i.e., + by simplifying and substituting the values of α + β and αβ. 1 Similarly find the product = αβ Frame the quadratic equation whose zeroes are and. 6. Frame a cubic polynomial whose zeroes are in Arithmetic Progression with a common difference 2 and the sum of the zeroes is 12.
Procedure: A cubic polynomial has at most 3 zeroes since the zeroes are in A.P.,with a common difference 2, all the three zeroes are distinct. Let the zeroes be a 2, a, a + 2 Equate the sum of the zeroes to 12 Simplify to get the values of the zeroes of the polynomial. Frame the cubic polynomial (x α) (x β) (x γ) and simplify to write it in the standard form. 7. Saritha said that her mother's age is 4 more than thrice her age and her father's age is 6 less than four times her age. Express the age of her mother and father as linear polynomials. Can you find Saritha's parents age if her father is four years older than her mother. Hint: Assume the age of Saritha as 'x'. Express her mother's age in terms of x. Express her father's age in terms of x. Find the age of Saritha from the equation (3x + 4) + 4 = 4x 6 Multiple Choice Questions 1. The graph of y = ax + b where a, b are real numbers and a 0 intersects X - axis at... ( b b b b A) (0, 0) B) 0, ) ( C), 0 ) ( D), ) a a a a Ans: C 2. Which of the following are not Polynomials? (i) 2x 3 (ii) (iii) 4z 2 + (iv) p -2 + 1 x 1 7 A) (i) & (ii) B) (ii), (iii) C) (iii & iv) D) (ii) & (iv) Ans: D Writer: V. Padma Priya
Polynomials 10 th class Mathematics Polynomials are algebraic expressions constructed using constants and variables. Coefficients operate on variables, which can be raised to various powers of non- negative integer exponents. 5x 3, -2x 2 + 3x + 2, 2y 3 are examples of polynomials. 3,,, 2x 3 are not polynomials as the exponents of the variables x-1 3 x y 3 are not non-negative integers. The highest power of x is p(x) is called the degree of the polynomial p(x). The polynomial of degree 1 is a linear polynomial. e.g.: 2x + 3, y - 5, 2x - 5 are linear polynomials. The polynomial of degree 2 is called a quadratic polynomial. 5 1 e.g.: 3x 2 + 2x + 1, x 2 - x + 2 are quadratic polynomials in variable x. 3 8 The polynomial of degree 3 is called a cubic polynomial. 2 1 e.g.: 8x 2-3x 3, 5x 3-3x 2 + 2x + 7, s 3 - s 2 + 2s + 4 are cubic 3 5 polynomials. p(x) = a 0 x n + a 1 x n-1 + a 2 x n-2 +...+ a n-1 x + a n is a polynomial of degree n, where a 0, a 1, a 2,..., a n are real coefficients and a 0 0. The general form of a first degree polynomial in variable x is ax + b where a, b are real numbers, a 0. The general form of a second degree polynomial in variable x is ax 2 + bx + c where a, b, c are real numbers and a 0. If p(x) is a polynomial in x, and if k is a real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k and is denoted by p(k). A real number k is said to be a zero of a polynomial p(x) if p(k) = 0 For a linear polynomial ax + b, a 0, the graph of y = ax + b is a straight line
-b which intersect the x-axis at exactly one point namely (, 0 ). a The linear polynomial ax + b, a 0 has exactly one zero, the x-coordinate of the point where the graph of y = ax + b intersects the x-axis. The graph of a quadratic polynomial is a curve called parabola. The zeroes of a quadratic polynomial ax 2 + bx + c, a 0 are the x-coordinates of the points where the parabola representing y = ax 2 + bx + c intersects the x-axis. The polynomial p(x) of degree 2 has atmost two zeroes. The polynomial p(x) of degree 3 has atmost three zeroes. In general, the polynomial p(x) of degree n has atmost n zeroes. If α and β are the zeroes of a quadratic polynomial ax 2 + bx + c, a 0 then -b c α + β = a, αβ = a If α, β, γ are the zeroes of a cubic polynomial ax 3 + bx 2 + cx + d, a 0 then -b c -d α + β + γ =, αβ + βγ + γα =, αβγ = a a a The division algorithm states that given any polynomial p(x) and any non - zero polynomial g(x), there are polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x) where either r(x) = 0 or degree of r(x) < degree of g(x) if r(x) 0 If q(x) is a linear polynomial then r(x) = r, a constant. If degree of q(x) = 1, then degree of p(x) = 1 + degree of g(x) If p(x) is divided by (x - a), then the remainder is p(a). If r = 0, we say q(x) divides p(x) exactly or q(x) is a factor of p(x). Writer : V. Padmapriya