Advertisements
Advertisements
प्रश्न
Show that every positive odd integer is of the form (4q +1) or (4q+3), where q is some integer.
उत्तर
Let a be a positive odd integer
Using the division algorithm on a and b = 4
a = 4q + r Since 0 ≤ r < 4, the possible remainders are 0,1, 2 and 3
∴ a can be 4q or 4q + 1 or 4q + 2 or 4q +3 , where q is the quotient
Since a is odd ,a cannot be 4q + 4q + 2
∴ Any odd integer is of the form 4q + 1 or 4q + 3 , where q is some integer.
APPEARS IN
संबंधित प्रश्न
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following : p(x) = x4 – 3x2 + 4x + 5, g(x) = x2 + 1 – x
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial
x3 – 3x + 1, x5 – 4x3 + x2 + 3x + 1
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm
deg p(x) = deg q(x)
Find all the zeros of the polynomial x3 + 3x2 − 2x − 6, if two of its zeros are `-sqrt2` and `sqrt2`
If (a-b) , a and (a + b) are zeros of the polynomial `2x^3-6x^2+5x-7` write the value of a.
State Division Algorithm for Polynomials.
Find the quotient and remainder of the following.
(8y3 – 16y2 + 16y – 15) ÷ (2y – 1)
The sum of (x + 5) observations is (x3 + 125). Find the mean of the observations
If on division of a non-zero polynomial p(x) by a polynomial g(x), the remainder is zero, what is the relation between the degrees of p(x) and g(x)?
Find k so that x2 + 2x + k is a factor of 2x4 + x3 – 14 x2 + 5x + 6. Also find all the zeroes of the two polynomials.
