true/false

1. a cubic polynomial has at least one zero..............
2. a quadratic polynomial an have at most two zeroes..........
3. if r(x)is the remainder and p(x) is the divisor, then degree r(x) < degree p(x)............
4. if zeroes of a quadratic polynomial ax^2+b^x+c are both negative then a,b and c all have the same sign..............
5. the quadratic polynomial x^2+kx+k can have equal zeroes for some odd integer k>1......................

someone plzzzz help me in these true and false ........................................

False

False

false

To determine whether the statements are true or false, we will go through each statement one by one.

1. A cubic polynomial has at least one zero.

This statement is true. According to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n zeros, counting both real and complex roots. Since a cubic polynomial has a degree of 3, it will have at least one zero.

2. A quadratic polynomial can have at most two zeroes.

This statement is also true. A quadratic polynomial is a polynomial of degree 2, which means it can have at most two zeros. However, it is possible for a quadratic polynomial to have only one or even no real zeros.

3. If r(x) is the remainder and p(x) is the divisor, then the degree of r(x) is less than the degree of p(x).

This statement is true. When dividing a polynomial p(x) (the dividend) by another polynomial q(x) (the divisor), the degree of the remainder r(x) is always less than the degree of the divisor q(x).

4. If the zeroes of a quadratic polynomial ax^2 + bx + c are both negative, then a, b, and c all have the same sign.

This statement is false. The sign of the coefficients a, b, and c is not determined by the sign of the zeros. In fact, the signs of a, b, and c can vary independently. To determine their signs, more information about the quadratic polynomial is needed.

5. The quadratic polynomial x^2 + kx + k can have equal zeroes for some odd integer k > 1.

This statement is true. For a quadratic polynomial to have equal roots (zeroes), the discriminant (b^2 - 4ac) must be equal to zero. In the given polynomial x^2 + kx + k, when we substitute k for both a and c, the discriminant becomes k^2 - 4k = k(k - 4). Setting this equal to zero, we have k(k - 4) = 0. This equation is satisfied when k = 0 or k = 4. However, the statement specifies an odd integer k > 1, so the only valid solution is k = 4.

So, the answers are:
1. True
2. True
3. True
4. False
5. True

True