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Write the standard form of a linear polynomial with real coefficients.
As we know the condition for linear polynomial degree equals to 1
So,
f(x) = ax1 + b,
Where,
a 0
Define a polynomial with real coefficients.
Define degree of a polynomial.
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.
The sum and product of the zeros of a quadratic polynomial are and — 3 respectively. What is the quadratic 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