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Can x² - 1 be the quotient on division of x⁶ + 2x³ + x - 1 by a polynomial in x of degree 5

Solution:

Given, the polynomial x⁶ + 2x³ + x - 1 is divided by a polynomial in x of degree 5.

We have to check whether the quotient is x² - 1.

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 r(x) = 0 or degree r(x) < degree g(x).

Polynomial in degree 5 because the degree of the product of the quotient and the divisor should be equal to the power of the dividend.

When a degree of polynomial 6 is divided by a degree of 5 polynomials, the quotient will be degree 1.

Here, p(x) = x⁶ + 2x³ + x - 1

g(x) = x in degree of 5

q(x) = x² - 1

So, (x⁶ + 2x³ + x - 1) = (x in degree of 5)(x² - 1) + r(x)

= x in degree of 7

Therefore, x² - 1 cannot be the quotient of p(x).

✦ Try This: Can x² - 1 be the quotient on division of 2x⁶ + 4x³ + 6x - 1 by a polynomial

in x of degree 5

Given, the polynomial 2x⁶ + 4x³ + 6x - 1 is divided by a polynomial in x of degree 5.

We have to check whether the quotient is x² - 1.

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 r(x) = 0 or degree r(x) < degree g(x).

Polynomial in degree 5 because the degree of the product of the quotient and the divisor should be equal to the power of the dividend.

When a degree of polynomial 6 is divided by a degree of 5 polynomials, the quotient will be degree 1.

Here, p(x) = 2x⁶ + 4x³ + 6x - 1

g(x) = x in degree of 5

q(x) = x² - 1

So, (2x⁶ + 4x³ + 6x - 1) = (x in degree of 5)(x² - 1) + r(x)

= x in degree of 7

Therefore, x² - 1 cannot be the quotient on division of x⁶ + 2x³ + x - 1 by a polynomial

in x of degree 5

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 2

NCERT Exemplar Class 10 Maths Exercise 2.2 Problem 1 (i)

Summary:

x² - 1 cannot be the quotient on division of x⁶ + 2x³ + x - 1 by a polynomial in x of degree 5

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