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Polynomials

For which values of a and b, are the zeroes of q(x) = x³ + 2x² + a also the zeroes of the polynomial p(x) = x⁵ – x⁴ – 4x³ + 3x² + 3x + b? Which zeroes of  p(x) are not the zeroes of q(x)?

Polynomials

Find k so that x² + 2x + k is a factor of 2x⁴  + x³ – 14x² + 5x + 6. Also find all the zeroes of the two polynomials.

Polynomials

Given that the zeroes of the cubic polynomial x³ – 6x² + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Polynomials

If the remainder on division of x³ + 2x² + kx +3 by x – 3 is 21, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x³ + 2x² + kx – 18.

Polynomials

Find the zeroes of the polynomial x² + 1/6x – 2, and verify the relation between the coefficients and the zeroes of the polynomial. 

Polynomials

Are the following statement ‘True’ or ‘False’? Justify your answer.The only value of k for which the quadratic polynomial kx² + x + k has  equal zeros is  1/2

Polynomials

Are the following statement ‘True’ or ‘False’? Justify your answer.

If all three zeroes of a cubic polynomial x³ + ax² – bx + c are positive, then at least one of a, b and c is non-negative.

Polynomials

Are the following statement ‘True’ or ‘False’? Justify your answer.

If  all  the  zeroes  of  a  cubic  polynomial  are  negative,  then  all  the coefficients and the constant term of the polynomial have the same sign.

Polynomials

Are the following statement ‘True’ or ‘False’? Justify your answer.

If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.

Polynomials

Are the following statement ‘True’ or ‘False’? Justify your answer.

If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.