Define a polynomial with real coefficients.

Define degree of a polynomial.

Write the standard form of a linear polynomial with real coefficients.

Write the standard form of a quadratic polynomial with real coefficients.

Write the standard form of a cubic polynomial with real coefficients.

Define the value of a polynomial at a point.

Define zero of a polynomial.

Write the family of quadratic polynomials having and 1 as its zeros.

If the product of zeros of the quadratic polynomial f(x) = x² — 4x + k is 3, find the value of k.

If the sum of the zeros of the quadratic polynomial f (x) = kx² — 3x + 5 is 1, write the value of k.

In Fig. 2.17, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.

The graph of a polynomial y = f (x), shown in Fig. 2.18. Find the number of real zeros of f (x).

The graph of the polynomial f(x) = ax² + bx + c is as shown below (Fig. 2.19). Write the signs of 'a' and b² – 4ac.

The graph of the polynomial f(x) = ax² + box + c is as shown in Fig. 2.20. Write the value of b² – 4ac and the number of real zeros of f (x).

In Q. No. 14, write the sign of c.

In Q. No. 15, write the sign of c.

The graph of a polynomial/ (x) is as shown in Fig. 2.21. Write the number of real zeros of f (x).

If x = 1 is a zero of the polynomial f (x) = x³ – 2x² + 4x + k, write the value of k.

State division algorithm for polynomials.

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) .q(x)+ r(x), where degree r (x) = 0.

Write a quadratic polynomial, sum of whose zeros is 2√3 and their product is 2.

If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.

If f(x) = x³ + x² – ax + b is divisible by x² – x write the values of a and b.

If a – b, a and a + b are zeros of the polynomial f (x) = 2x³ – 6x² + 5x – 7, write the value of a.

Write the coefficients of the polynomial p(z) = z⁵ – 2z² + 4.

Write the zeros of the polynomial x² – x – 6.

If (x + a) is a factor of 2x² + 2ax + 5x + 10, find a.

For what value of k, – 4 is a zero of the polynomial x² – x – (2k + 2)?

If 1 is a zero of the polynomial p(x) = ax² – 3(a – 1) x – 1, then find the value of a.

If α, β are the zeros of a polynomial such that α + β = – 6 and αβ = – 4, then write the polynomial.

If α, β are the zeros of the polynomial 2y² + 7y + 5, write the value of α + β + αβ.

For what value of k, is 3 a zero of the polynomial 2x² + x + k?

For what value of k, is – 3 a zero of the polynomial x² + 11x + k?

For what value of k, is – 2 a zero of the polynomial 3x² + 4x + 2k?

If a quadratic polynomial f (x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f(x)?

If a quadratic polynomial f (x) is a square of a linear polynomial, then its two zeroes are coincident. (True/False)

If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True/False)

If f(x) is a polynomial such that f(a)f(b)< 0, then what is the number of zeros lying between a and b?

If graph of quadratic polynomial ax² + bx + c cuts positive direction of y-axis, then what is the sign of c?

If the graph of quadratic polynomial ax² + bx + c cuts negative direction of y-axis, then what is the sign of c?

If α, β are the zeros of the polynomial f (x) = x² + x + 1, then =

If α, β are the zeros of the polynomial p(x) = 4x² + 3x + 7, then =

If one zero of the polynomial f (x) = (k² + 4) x² + 13x + 4k is reciprocal of the other, then k =

If the sum of the zeros of the polynomial f(x) = 2x³ – 3kx² + 4x – 5 is 6, then the value of k is

If α and β are the zeros of the polynomial f(x) = x² + px + q, then a polynomial having and is its zeros is

If α, β are the zeros of polynomial f(x) = x² – p (x + 1) – c, then (α + 1) (β + 1) =

If α, β are the zeros of the polynomial f(x) = x² – p (x + 1) – c such that (α + 1) (β + 1) = 0, then c =

If f(x) = ax² + bx + c has no real zeros and a + b + c < 0, then

If the diagram in Fig. 2.22 shows the graph of the polynomial f (x) = ax² + bx + c, then

Figure 2.23 shows the graph of the polynomial f(x) = ax² + bx + c for which

If the product of zeros of the polynomial f(x) = ax³ –6x² + 11x – 6 is 4, then a =

If zeros of the polynomial f (x) = x³ – 3px² + qx – r are in A.P., then

If the product of two zeros of the polynomial f (x) = 2xZ³ + 6x² – 4x + 9 is 3, then its third zero is

If the polynomial f(x) = ax³ + bx – c is divisible by the polynomial g(x) = x² + bx + c, then ab =

In Q. No. 14, c =

If one root of the polynomial f (x) = 5x² + 13x + k is reciprocal of the other, then the value of k is

If α, β, γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, then =

If α, β, γ are the zeros of the polynomial f(x) = ax³ + bx² + cx + d, then α² + β² + γ² =

If α, β, γ are the zeros of the polynomial f(x) = x³ – px² + qx – r, then =

If α, β are the zeros of the polynomial f(x) = ax² + bx + c, then =

If two of the zeros of the cubic polynomial ax³ + bx² + cx + d are each equal to zero, then the third zero is

If two zeros of x³ + x² – 5x – 5 are √5 and –√5 , then its third zero is

The product of the zeros of x³ + 4x² + x – 6 is

What should be added to the polynomial x² – 5x + 4, so that 3 is the zero of the resulting polynomial?

What should be subtracted to the polynomial x² – 16x + 30, so that 15 is the zero of the resulting polynomial?

A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is

If two zeroes of the polynomial x³ + x² – 9x – 9 are 3 and – 3, then its third zero is

If √5 and – √5 are two zeroes of the polynomial x³ + 3x² – 5x – 15, then its third zero is

If x + 2 is a factor of x² + ax + 2b and a + b = 4, then

The polynomial which when divided by – x² + x – 1 gives a quotient x – 2 and remainder 3, is