Listen NCERT Audio Books to boost your productivity and retention power by 2X.

Find the quotient and the remainder when:

f(x) = x^{4} – 3x^{2} + 4x + 5 is divided by g(x) = x^{2} + 1 – x.

It is given in the question that,

f (x) = x^{4} – 3x^{2} + 4x + 5

And, g (x) = x^{2} + 1 – x

Hence,

Quotient q (x) = x^{2} + x - 3

Remainder r (x) = 8

Verify that 3, ‒2, 1 are the zeros of the cubic polynomial p(x) = x^{3} – 2x^{2} – 5x + 6 and verify the relation between its zeros and coefficients.

Verify that 5, –2 and are the zeros of the cubic polynomial p(x) = 3x^{3} - 10x^{2}– 27x + 10 and verify the relation between its zeros and coefficients

Find a cubic polynomial whose zeros are 2, –3 and 4

Find a cubic polynomial whose zeros are , 1 and –3.

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, –2 and –24 respectively.

f(x) = x^{3} – 3x^{2} + 5x –3 is divided by g(x) = x^{2} – 2.

f(x) = x^{4} – 5x + 6 is divided by g(x) = 2 – x^{2}.

By actual division, show that x^{3} – 3 is a factor 2x^{4} + 3x^{3} – 2x^{2} – 9x – 12.

On dividing 3x^{3} + x^{2} + 2x + 5 by a polynomial g(X), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).

Verify division algorithm for the polynomials f(x) = 8 + 20x + x^{2} ‒ 6x^{3} and g(x) = 2 + 5x ‒ 3x^{2}.

It is given that ‒1 is one of the zeros of the polynomial x^{3} + 2x^{2} ‒ 11x ‒ 12. Find all the zeros of the given polynomial.

If 1 and ‒2 are two zeros of the polynomial (x^{3} ‒ 4x^{2} ‒ 7x + 10), find its third zero.

If 3 and ‒3 are two zeros of the polynomial (x^{4} + x^{3} ‒ 11x^{2} ‒ 9x + 18), find all the zeros of the given polynomial.

If 2 and ‒2 are two zeros of the polynomial (X^{4} + x^{3} ‒ 34x^{2} ‒ 4x + 120), find all the zeros of the given polynomial.

Find all the zeros of (x^{4} + x^{3} ‒ 23x^{2} ‒ 3x + 60), if it is given that two of its zeros are √3 and ‒√3.

Find all the zeros of (2x^{4} ‒ 3x^{3} ‒ 5x^{2} + 9x ‒ 3), it being given that two of its zeros are √3 and – √3.

Obtain all other zeros of (x^{4} + 4x^{3} ‒ 2x^{2} ‒ 20x ‒15) if two of its zeros are √5 and – √5.

Find all the zeros of the polynomial (2x^{4} ‒ 11x^{3} + 7x^{2} + 13x ‒ 7), it being given that two of its zeros are (3 + √3) and (3 ‒ √3)