A polynomial of degree 4, with real coefficients, can have at most how many real roots?

**Question 50 (Final Redraft):** If the zeros of the quadratic polynomial P(x) = (k-1)x² - 5x + (2k+1) are reciprocal of each other, then the value of k is:

**Question (Redrafted and Corrected):** If the zeros of the quadratic polynomial kx² + 2x - 3k are equal in magnitude but opposite in sign, then the value of k is:

If the zeros of the quadratic polynomial kx² + 2x - 3k are equal in magnitude but opposite in sign, then the value of k is:

How many real zeros does the polynomial P(x) = (x² - 4)(x + 1)² have?

For a quadratic polynomial P(x) = ax² + bx + c, the vertex of its parabolic graph is at x =:

If x = 0 is a zero of the polynomial P(x) = ax² + bx + c, then:

The constant term of a polynomial P(x) can be found by evaluating:

If alpha and beta are the zeros of the polynomial P(x) = x² + x - 2, then (1/alpha + 1/beta) is equal to:

If P(x) is a polynomial such that P(x) = P(-x) for all x, then P(x) is an:

If a polynomial has real coefficients, its complex non-real zeros always occur in:

The maximum number of terms in a cubic polynomial is:

If alpha and beta are the zeros of a polynomial P(x) such that alpha + beta = 5 and alpha * beta = 6, then P(x) is:

Which of the following statements about polynomial P(x) = x² + 1 is true?

Let's try: If the product of the zeros of P(x) = kx² + 6x + 8 is 4, then the value of k is:

If the product of the zeros of P(x) = x² + mx - 16 is 4, then the value of m is:

If the product of the zeros of P(x) = x² + mx + 16 is 4, then the value of m is:

The number of real zeros of the polynomial P(x) = (x² + 1)(x - 3) is:

Which of the following cannot be the graph of a quadratic polynomial?

If a and b are the zeros of the polynomial x² + px + q, and if a+b = ab, then which of the following is true?

If one zero of the polynomial P(x) = ax² + bx + c is '0', then:

The graph of a linear polynomial P(x) = ax + b (where a != 0) is always a:

For a quadratic polynomial P(x) = ax² + bx + c, if b² - 4ac = 0, then the polynomial has:

What is the maximum number of terms in a polynomial of degree 'n' if all coefficients are non-zero?

If one zero of the polynomial P(x) = x³ + 3x² - x - 3 is 1, find the other two zeros.

If a real number 'a' is a zero of a polynomial P(x), then which of the following is always true?

If alpha and beta are the zeros of the polynomial P(x) = x² - 8x + 16, what is the value of P(alpha + beta)?

How many terms are there in the polynomial 5x⁴ - 2x³ + 7x² - x + 10?

Which of the following is NOT a polynomial in one variable?

If the sum of the zeros of the polynomial P(x) = x² - 2x + (2k-1) is equal to their product, then the value of k is:

Which of the following polynomials could have zeros -1/2, 1, and 2?

If the polynomial P(x) = x³ + 4x² - 3x - 18 is exactly divisible by (x - 2), then which of the following is also a factor?

If the quadratic polynomial P(x) = kx² + 4x + 1 has equal real zeros, then k must be:

A quadratic polynomial P(x) has no real zeros. Which of the following is true about its graph?

If alpha and beta are the zeros of the polynomial 2x² + 5x + k, and alpha² + beta² + alpha*beta = 21/4, what is the value of k?

The degree of the zero polynomial (P(x) = 0) is:

If the zeros of a quadratic polynomial are (2+sqrt(3)) and (2-sqrt(3)), then the polynomial is:

A quadratic polynomial whose zeros are 5 and -2 is:

When x³ - 2x² + kx + 5 is divided by (x-2), the remainder is 1. Find the value of k.

If alpha and beta are the zeros of the polynomial P(x) = 2x² + 7x + 3, what is the value of alpha² + beta²?

If sqrt(5) is a zero of the polynomial f(x) = x² - 5, what is its other zero?

If the graph of a quadratic polynomial ax² + bx + c opens downwards, which of the following is definitely true?

A polynomial of degree 4 can have at most how many points of intersection with the x-axis?

What is the degree of the polynomial P(x) = (x² + 1)(x - 2) - (x³ + 5x)?

If (x-1) and (x+1) are factors of P(x) = ax³ + bx² + cx + d, then which of the following statements must be true?

If one zero of the cubic polynomial P(x) = x³ + ax² + bx + c is -1, and the product of the other two zeros is 3, what is the value of 'c'?

If x = -1 is a zero of the polynomial P(x) = ax³ + bx² + cx + d, then which of the following is true?

What is the minimum number of real roots a cubic polynomial can have?

What is the maximum number of real roots a cubic polynomial can have?

If the sum of two zeros of the polynomial P(x) = x³ - 5x² + kx - 7 is 3, what is the value of k?

What is the remainder when x³ - ax² + 6x - a is divided by (x - a)?

For a quadratic polynomial ax² + bx + c whose graph opens upwards and intersects the x-axis at two distinct points, which of the following is true?

The number of polynomials having zeros -2 and 5 is:

If alpha and beta are the zeros of the quadratic polynomial P(x) = x² - 6x + 4, find a quadratic polynomial whose zeros are 3alpha and 3beta.

If (x+a) is a factor of the polynomial 2x² + 2ax + 5x + 10, then the value of a is:

If a zero of the polynomial f(x) = x² - 5x + 2k is 2, then the value of k is:

A polynomial of degree 'n' has at most:

Assertion (A): If P(x) = ax² + bx + c has real zeros, then b² - 4ac >= 0. Reason (R): The discriminant (b² - 4ac) determines the nature of the zeros of a quadratic polynomial.

If the sum of the zeros of the polynomial f(x) = 2x² - 3kx + 5 is 3, what is the value of k?

If alpha and beta are the zeros of the polynomial P(x) = x² - x - 4, what is the value of (1/alpha + 1/beta - alpha*beta)?

If two zeros of the polynomial P(x) = x⁴ - 6x³ - 26x² + 138x - 35 are (2 + sqrt(3)) and (2 - sqrt(3)), find the other two zeros.

Which of the following expressions is NOT a polynomial?

If one zero of the quadratic polynomial P(x) = (k²+4)x² + 13x + 4k is the reciprocal of the other, find the positive value of k.

If alpha, beta, gamma are the zeros of the polynomial P(x) = x³ - 6x² + 11x - 6, then what is the value of (alpha*beta + beta*gamma + gamma*alpha)?

If alpha and beta are the zeros of the polynomial f(x) = x² - px + q, find a polynomial whose zeros are 1/alpha and 1/beta.

When a polynomial P(x) = x³ - 3x² + x + 2 is divided by a polynomial G(x), the quotient is x - 2 and the remainder is -2x + 4. What is G(x)?

If alpha and beta are the zeros of the quadratic polynomial P(x) = x² - 5x + k, and alpha - beta = 1, what is the value of k?

What is the degree of the polynomial P(x) = (x² + 3x + 1)(2x² - x + 5)?

If one zero of the polynomial p(x) = x² - (a+1)x + 5 is 5, then the other zero is:

The number of real zeros of the polynomial f(x) = (x-1)² + 1 is:

If the zeros of the polynomial x² + px + q are reciprocal of each other, then:

Which of the following polynomials is a linear polynomial?

If the product of two zeros of the polynomial x³ - 5x² - 2x + 24 is 12, then the third zero is:

If two zeros of the polynomial x⁴ - 6x³ - 26x² + 138x - 35 are (2 + sqrt(3)) and (2 - sqrt(3)), then the other two zeros are:

Assertion (A): If the zeros of the polynomial x² + px + q are twice the zeros of 2x² - 5x - 3, then p = 5 and q = -18. Reason (R): For a quadratic polynomial ax² + bx + c, sum of zeros = -b/a and product of zeros = c/a.

The sum of the zeros of the polynomial x² + 2x - 8 is:

If x² - 3 is a factor of x⁴ - 2x³ - 5x² + 6x + 9, then the other factors are:

The maximum number of times a polynomial of degree n can intersect the x-axis is:

How many zeros does the polynomial x² + 1 have?

If alpha and beta are the zeros of the polynomial f(x) = x² - (p+1)x + 5, and alpha + beta = alpha * beta, then the value of p is:

If alpha and beta are the zeros of the polynomial 2x² - x - 6, then a polynomial whose zeros are 1/alpha and 1/beta is:

The graph of a linear polynomial ax+b is a straight line which intersects the x-axis at:

What is the common zero of the polynomials x² - 4 and x² + x - 6?

The sum of the zeros of the cubic polynomial 2x³ - x² + 5x - 7 is:

If one zero of the polynomial p(x) = x² + (3k+1)x + k is 2, then the value of k is:

If alpha and beta are the zeros of the quadratic polynomial x² - 3x + 2, then the value of (alpha + beta)² - 2alpha*beta is:

If the zeros of the quadratic polynomial x² + (a+1)x + b are 2 and -3, then:

The number of polynomials whose zeros are 3 and -4 is:

If alpha, beta, gamma are the zeros of the polynomial 3x³ - 5x² - 11x - 3, then alpha*beta*gamma is:

If x+1 is a factor of the polynomial x³ + kx² - x + 2, then the value of k is:

If the polynomial x⁴ + ax³ + 2x² + bx + 1 is divided by x-1 and x+1, the remainders are 5 and 7 respectively. The values of a and b are:

The number of zeros of the polynomial whose graph is parallel to the x-axis and passes through (0, 3) is:

If alpha and beta are the zeros of the polynomial P(x) = x² - 4x + 3, then a polynomial whose zeros are 1/alpha and 1/beta is:

A quadratic polynomial has at most:

If alpha and beta are the zeros of the polynomial x² + 7x + 10, then the value of 1/alpha + 1/beta is:

If the zeros of the quadratic polynomial ax² + bx + c are equal, then:

The value of the polynomial 2x³ - 3x² + 4x - 5 at x = 1 is:

If the polynomial x³ - x² - ax + b is divisible by (x-1) and (x+1), the values of a and b are:

If the polynomial x³ + x² - ax + b is divisible by (x-1) and (x+1), the values of a and b are:

If alpha and beta are zeros of the polynomial f(x) = x² - 8x + k, such that alpha² + beta² = 40, then k = ?

A quadratic polynomial, one of whose zeros is 2 - sqrt(5) and the sum of its zeros is 4, is:

If alpha and beta are the zeros of x² - 5x + 6, then the polynomial whose zeros are alpha+1 and beta+1 is:

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

If alpha and beta are the zeros of the polynomial P(x) = x² - 6x + a, and 3alpha + 2beta = 16, then the value of 'a' is:

If alpha and beta are the zeros of the polynomial P(x) = x² - 6x + a, and 3alpha + 2beta = 20, then the value of 'a' is:

If one zero of the polynomial x² - 4x + 1 is (2 + sqrt(3)), then the other zero is:

What is the degree of the polynomial 7x³ - 3x⁵ + 2x² + 9?

The product of the zeros of the quadratic polynomial 3x² + 5x - 2 is:

If the square of the difference of the zeros of the quadratic polynomial x² + px + 45 is 144, then the value of p is:

Consider the polynomial P(x) = (x-1)(x+2) and Q(x) = (x-2)(x+1). Which of the following statements is true about their zeros?

If alpha, beta, gamma are the zeros of the cubic polynomial x³ - 3x² + x + 1, then (alpha*beta + beta*gamma + gamma*alpha) is:

The minimum number of zeros a cubic polynomial can have is:

If the polynomial x² + x + k has a zero at x=2, then the value of k is:

If alpha and beta are the zeros of the polynomial 2x² + 7x - 3, then the value of (alpha + beta) / (alpha * beta) is:

If a and b are the zeros of the polynomial x² - 1, then a polynomial whose zeros are 2a/b and 2b/a is:

If (x-2) is a factor of the polynomial x² - ax + b, and a+b=11, then the values of a and b are:

If (x+1) is a factor of x² - ax - 2b, and 2a + 3b = 4, then the values of a and b are:

The degree of the polynomial 0x⁵ + 2x² + 3x + 1 is:

If alpha and beta are the zeros of the polynomial f(x) = ax² + bx + c, then (1/alpha + 1/beta)² is equal to:

If x² + x - p has a zero at x= -2, then the value of p is:

The number of real zeros for the polynomial x² + 4 is:

Which of the following is a zero of the polynomial x³ - 3x² + 4x - 12?

If alpha and beta are the zeros of 2x² + 5x - 9, then the value of alpha² + beta² is:

The remainder when x³ + 1 is divided by x + 1 is:

The product of the zeros of the polynomial P(x) = ax³ + bx² + cx + d is:

If alpha and beta are the zeros of the quadratic polynomial P(x) = x² - (k-6)x + (2k+1), and alpha + beta = alpha * beta, then the value of k is:

Which of the following polynomials has -3 as a zero?

If a polynomial P(x) is divided by x - 1, the remainder is 5. If P(x) is divided by x - 2, the remainder is 10. Then the remainder when P(x) is divided by (x - 1)(x - 2) is:

The degree of the polynomial (x-1)(x-2)(x-3)(x-4) is:

A quadratic polynomial, whose sum and product of zeros are 3 and -10 respectively, is:

If alpha and beta are the zeros of the polynomial f(x) = x² - 5x + k, and alpha - beta = 1, then the value of k is:

The value of 'm' for which the polynomial x² - (m+4)x + 6 has a zero at x=3 is:

The value of 'm' for which the polynomial x² - (m+2)x + 6 has a zero at x=3 is:

If one of the zeros of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it:

If the sum of the zeros of the polynomial 3x² - (k+6)x + 2k - 1 is -1, then the value of k is:

If the sum of the zeros of the polynomial 3x² - (k+6)x + 2k - 1 is -3, then the value of k is:

For a quadratic polynomial ax² + bx + c, if a and c have opposite signs, then the zeros of the polynomial are:

If alpha and beta are the zeros of the quadratic polynomial f(x) = x² - px + q, then a polynomial having 1/alpha and 1/beta as its zeros is:

If two zeros of the polynomial x³ - 4x² - 3x + 12 are sqrt(3) and -sqrt(3), then its third zero is:

The value of the polynomial p(x) = 5x - 4x² + 3 when x = -1 is:

If -1 is a zero of the polynomial p(x) = ax³ - x² + 5x + 1, then the value of a is:

A polynomial of degree n has:

If alpha and beta are zeros of x² + x + 1, then the polynomial whose zeros are 1/alpha and 1/beta is:

What should be added to the polynomial x² - 5x + 4 so that (x - 3) becomes its factor?

If alpha and beta are the zeros of the quadratic polynomial f(x) = ax² + bx + c, then alpha/beta + beta/alpha is equal to:

The number of polynomials having zeros as -2 and 5 is:

If x+a is a factor of 2x² + 2ax + 5x + 10, then the value of a is:

Assertion (A): The polynomial x⁴ + 4x² + 5 has two real zeros. Reason (R): The degree of a polynomial is the number of its real zeros.

If alpha, beta, gamma are the zeros of the polynomial 2x³ + x² - 13x + 6, then alpha*beta + beta*gamma + gamma*alpha is equal to:

If the polynomial x⁴ + 2x³ + 8x² + 12x + 18 is divided by x² + 5, the remainder comes out to be px + q. The values of p and q are:

If two of the zeros of a cubic polynomial ax³ + bx² + cx + d are 0, then the third zero is:

If alpha and beta are the zeros of the polynomial f(x) = x² - 8x + k, such that alpha² + beta² = 40, then the value of k is:

If the sum of the zeros of the quadratic polynomial kx² + 2x + 3k is equal to their product, then k is equal to:

If x = 2 and x = 3 are zeros of the polynomial 3x² - 2kx + 2m, then the values of k and m are respectively:

If alpha and beta are the zeros of the polynomial x² + 6x + 2, then the value of (1/alpha + 1/beta) is:

A quadratic polynomial, whose zeros are -2 and 5, is:

If one of the zeros of the quadratic polynomial (k-1)x² + kx + 1 is -3, then the value of k is:

If the product of two zeros of the polynomial p(x) = x³ - 6x² + 11x - 6 is 2, then the third zero is:

The maximum number of zeros a cubic polynomial can have is:

If alpha and beta are the zeros of the quadratic polynomial f(x) = x² - p(x+1) - c, then (alpha + 1)(beta + 1) is equal to:

The graph of y = p(x) is shown below. How many zeros does the polynomial have? (Assume the graph extends infinitely in the shown pattern)

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

Factorize a⁴ - b⁴.

The expanded form of (2x + 1)³ is:

If x - k is a factor of x² + 5x - 6, and k is a positive integer, then k is:

When x⁵ - 4x³ + 2x + 1 is divided by x - 2, the remainder is:

If (x - 1) is a factor of p(x) = x² + x + k, then the value of k is:

If p(x) = x³ + 2x² - 5x - 6, find p(-1).

The degree of the polynomial x⁵ - x⁴ + x³ - x² + x - 1 is:

Which of the following is a constant polynomial?

Factorize 3x² - x - 4.

If a + b = 10 and a² + b² = 58, find ab.

Which of the following is an identity?

If x + 2 is a factor of x³ + kx² + 3x + 6, then k is:

When x³ - 2x² + x + 1 is divided by x + 2, the remainder is:

How many zeroes does a linear polynomial have?

If p(x) = ax - 3 and p(1) = 0, then a is:

The degree of the polynomial 2x² - 3x⁴ + 7 is:

Which of the following is a monomial with a variable?

Factorize 12x² - 7x + 1.

Without actually calculating the cubes, find the value of (-12)³ + (7)³ + (5)³.

If x + y + z = 0, then x³ + y³ + z³ is equal to:

If x - 1 is a factor of 2x² + kx, then k is:

When x³ - ax² + 6x - a is divided by x - a, the remainder is:

Which of the following is NOT a zero of the polynomial x² - 4?

The degree of a non-zero constant polynomial is:

If p(x) = x² - 4, then p(2) is:

Factorize 8x³ - y³.

Factorize x³ + 6x² + 12x + 8.

Factorize y² - 5y + 6.

Factorize 49a² - 36b².

The value of 99² is:

The value of 103 x 107 using an identity is:

If a - b = 3 and ab = 4, find a² + b².

If x + y = 5 and xy = 6, find x² + y².

If x - a is a factor of p(x), then which of the following is true?

What is the remainder when 4x³ - 3x² + 2x - 1 is divided by x - 1?

A polynomial can have more than one zero. Is this statement true or false?

If p(x) = x³ - x² + x - 1, find p(1).

The coefficient of y³ in the polynomial 2y² - 7y³ + 5 is:

What is the degree of the polynomial sqrt(3)x⁴ - 5x + 2?

The term in a polynomial that does not contain any variable is called a:

Factorize 5x² - 20x.

Factorize x² - x - 6.

Factorize 27x³ + y³.

Factorize a³ - 8.

Expand (y - 3)³.

Expand (x + 2)³.

Expand (x - 2y + 3z)².

(a + b + c)² is equal to:

If x + 1 is a factor of ax³ + x² - 2x + 4a - 9, then a is equal to:

When x¹00 + 1 is divided by x - 1, the remainder is:

If x = -2 is a zero of the polynomial p(x) = x + a, then a is:

If p(y) = y³ - 1, find p(-1).

The degree of the polynomial 0x⁵ + 2x³ - x + 8 is:

What is the coefficient of x in the polynomial 5x³ + x² - 2x + 7?

A polynomial p(x) = cx + d, where c is not equal to 0, is a:

Which of the following is NOT a polynomial?

How many zeroes can a non-zero constant polynomial have?

The remainder when x⁴ + x³ - 2x² + x + 1 is divided by x - 1 is:

Factorize 2x² + 7x + 3.

Factorize x² + 5x + 6.

Expand (x + 2)(x + 5).

Factorize x² - 16.

Expand (2a - b)².

Expand (x + 3)².

For what value of k is x - 1 a factor of 4x³ + 3x² - 4x + k?

If x - 2 is a factor of x² - kx + 6, then the value of k is:

The remainder when x² + 2x + 1 is divided by x - 1 is:

When x³ + 1 is divided by x + 1, the remainder is:

Find the zero of the polynomial p(x) = 2x + 6.

What is the zero of the polynomial p(x) = x - 5?

If p(x) = 3x - 5, find p(2).

If p(x) = x² - 2x + 1, find p(0).

What is the degree of the zero polynomial?

What is the degree of the polynomial 7y⁵ - 3y² + 4?

What is the coefficient of x² in the polynomial 3x³ - 2x² + 5x - 1?

A polynomial with three terms is called a:

A polynomial of degree 1 is called a:

Which of the following expressions is a polynomial?

What is the degree of the polynomial 5?

If (x – 1/x) = 4, then evaluate (x² + 1/x²) and (x⁴ + 1/x⁴).

Factorise 64m³ – 343n³

Factorise 8a³ + b³ + 12a²b + 6ab²

Without actual division, prove that 2x^4 – 5x^3 + 2x^2 – x + 2 is divisible by x^2 – 3x + 2.

Evaluate (102)^3 using a suitable identity.

Factorise x^2 – 1 – 2a – a^2.

Check whether (7 + 3x) is a factor of (3x^3 + 7x).

Find the values of a and b so that (2x^3 + a x^2 + x + b) has (x + 2) and (2x – 1) as factors.

Find the value of x^3 + y^3 + z^3 – 3xyz if x + y + z = 15 and x^2 + y^2 + z^2 = 83

Calculate the perimeter of a rectangle whose area is 25x^2 - 35x + 12

Give an example of a monomial and a binomial having degrees of 82 and 99, respectively.

The value of 104 × 96 is

The value of 5.63 × 5.63 + 11.26 × 2.37 + 2.37 × 2.37 is

If one of the factor of x² + x – 20 is (x + 5). Find the other

If x + 2 is a factor of x³ – 2ax² + 16, then value of a is

Using a suitable identity, determine the value of (17)³ + (-12)³ + (-5)³

Find the product: (x – 3y) (x + 3y) (x² + 9y²)

Factorise: (a – b)³ + (b – c)³ + (c – a)³

If x + y = 12 and xy = 32, Find the value of x² + y²

Find the value of x³ + y³ + z³ – 3xyz if x² + y² + z² = 83 and x + y + z = 15

Find the value of 9x² + 4y² if xy = 6 and 3x + 2y = 12.

Find a quadratic polynomial, the sum and product of whose zeroes are √2 and -3/2, respectively. Also find its zeroes.

α and β are zeroes of the quadratic polynomial x² – 6x + y. Find the value of ‘y’ if 3α + 2β = 20

Find the quadratic polynomial if its zeroes are 0 and √5.

Find the zeroes of the polynomial 4x^2 – 4x – 8. Also, establish a relationship between the zeroes and coefficients.

Find the value of "p" from the polynomial x² + 3x + p, if one of the zeroes of the polynomial is 2.

If the product of zeroes of the polynomial p(x)=3x² + kx − 2 is 2/3 ​, find the value of k.

If the zeroes of the quadratic polynomial p(x) = ax² + bx + c are reciprocal of each other, prove that c = a.

Find a quadratic polynomial whose zeroes are 5 and −3.

Find the zeroes of the polynomial: p(x) = x² − 7x + 10 and verify the relation between zeroes and coefficients.

If one zero of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, find the value of a.