Advertisements
Advertisements
Question
Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5?
Solution
No, x2 – 1 cannot be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5.
Justification:
When a degree 6 polynomial is divided by degree 5 polynomial,
The quotient will be of degree 1.
Assume that (x2 – 1) divides the degree 6 polynomial with and the quotient obtained is degree 5 polynomial (1)
According to our assumption,
(Degree 6 polynomial) = (x2 – 1)(Degree 5 polynomial) + r(x) .....[Since, (a = bq + r)]
= (Degree 7 polynomial) + r(x) ......[Since, (x2 term × x5 term = x7 term)]
= (Degree 7 polynomial)
From the above equation, it is clear that, our assumption is contradicted.
x2 – 1 cannot be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5
Hence Proved.
APPEARS IN
RELATED QUESTIONS
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following
p(x) = x3 – 3x2 + 5x – 3, g(x) = x2 – 2
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following
p(x) = x4 – 5x + 6, g(x) = 2 – x2
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial
x2 + 3x + 1, 3x4 + 5x3 – 7x2 + 2x + 2
Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm
deg q(x) = deg r(x)
Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following f(x) = 4x3 + 8x2 + 8x + 7, g(x) = 2x2 − x + 1
Obtain all zeros of f(x) = x3 + 13x2 + 32x + 20, if one of its zeros is −2.
Find all the zeros of the polynomial x4 + x3 − 34x2 − 4x + 120, if two of its zeros are 2 and −2.
Find the quotient and remainder of the following.
(8y3 – 16y2 + 16y – 15) ÷ (2y – 1)
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.
Given that `x - sqrt(5)` is a factor of the cubic polynomial `x^3 - 3sqrt(5)x^2 + 13x - 3sqrt(5)`, find all the zeroes of the polynomial.
