If a polynomial p(x), has zeros at z1, z2, z3, ...
then p(x) is a multiple of (x - z1)*(x - z2)*(x - z3)...
To get the exact form of p(x) you also need to know the order of each root. If zk has order n then the relevant factor in p(x) is (x - zk)n
A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.
If the cubic polynomial you are given does not have an obvious factorization, then you must use synthetic division. I'm sure wikipedia can tell you all about that.
To find the quadratic polynomial whose zeros are 2 and -3, we can use the fact that a polynomial can be expressed in factored form as ( f(x) = a(x - r_1)(x - r_2) ), where ( r_1 ) and ( r_2 ) are the zeros. Here, substituting ( r_1 = 2 ) and ( r_2 = -3 ), we have ( f(x) = a(x - 2)(x + 3) ). Expanding this, we get ( f(x) = a(x^2 + x - 6) ). For simplicity, we can choose ( a = 1 ), giving us the polynomial ( f(x) = x^2 + x - 6 ).
Yes, a polynomial can have no rational zeros while still having real zeros. This occurs, for example, in the case of a polynomial like (x^2 - 2), which has real zeros ((\sqrt{2}) and (-\sqrt{2})) but no rational zeros. According to the Rational Root Theorem, any rational root must be a factor of the constant term, and if none exist among the possible candidates, the polynomial can still have irrational real roots.
A polynomial of degree ( n ) can have at most ( n ) real zeros. This is a consequence of the Fundamental Theorem of Algebra, which states that a polynomial of degree ( n ) has exactly ( n ) roots in the complex number system, counting multiplicities. Therefore, while all roots can be real, the maximum number of distinct real zeros a polynomial can possess is ( n ).
when the equation is equal to zero. . .:)
A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.
Polynomial fuction in standard form with the given zeros
by synthetic division and quadratic equation
Multiply x3 - 2x2 - 13x - 10
If the cubic polynomial you are given does not have an obvious factorization, then you must use synthetic division. I'm sure wikipedia can tell you all about that.
The zeros of a polynomial represent the points at which the graph crosses (or touches) the x-axis.
To find the quadratic polynomial whose zeros are 2 and -3, we can use the fact that a polynomial can be expressed in factored form as ( f(x) = a(x - r_1)(x - r_2) ), where ( r_1 ) and ( r_2 ) are the zeros. Here, substituting ( r_1 = 2 ) and ( r_2 = -3 ), we have ( f(x) = a(x - 2)(x + 3) ). Expanding this, we get ( f(x) = a(x^2 + x - 6) ). For simplicity, we can choose ( a = 1 ), giving us the polynomial ( f(x) = x^2 + x - 6 ).
The polynomial is (x + 1)*(x + 1)*(x - 1) = x3 + x2 - x - 1
Since no polynomial was given, no answer will be given.
x2 + 15x +36
The values of the variables which make the polynomial equal to zero