You need to multiply three terms, one for each zero. To have only two zeros, the polynomial would need to have a "double zero" (or more generally, a "multiple zero), that is, a repeated factor. In this case, the zeros can be one of the following:
-1, -1, 1, with the corresponding factors: (x+1)(x+1)(x-1)
or:
-1, 1, 1, with the corresponding factors: (x+1)(x-1)(x-1)
If you like, you can multiply these factors out to get the polynomial in standard form.
To find the polynomial degree you have only to add the exponents of all of the different components of the polynomial. In your case, you would add 1 and 5 from 4ab5 to get 6, 1 and 1 from 2ab to get 2, and 4 and 3 from 3a4b3 to get 7. Since the degree of the third component is the highest, that is you're answer.
Yes - but only if the domain is restricted. Normally the domain is the whole of the real numbers and over that domain it must have at least one real zero.
The degree of a polynomial function is the highest power any single term is raised to. For example, (5a - 2b^2) is a second degree function because the "b^2" is raised to the second power and the "a" is only raised to the (implied) first power. For (24xy-xy^3 + x^2) it is a third degree polynomial because the highest power is the cube of -xy.
It is the Coefficient. It only refers to the given term that it is front. e.g. 2x^2 - 3x + 1 The '2' in front of 'x^2' only refers to 'x^2'. The '-3' in front of 'x' is the coefficient of '-3' The '1' is a constant.
5
No. A polynomial can have as many degrees as you like.
To find the polynomial degree you have only to add the exponents of all of the different components of the polynomial. In your case, you would add 1 and 5 from 4ab5 to get 6, 1 and 1 from 2ab to get 2, and 4 and 3 from 3a4b3 to get 7. Since the degree of the third component is the highest, that is you're answer.
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial has a factor if and only if
In algebra, the factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.The factor theorem states that a polynomial has a factor if and only if
Yes - but only if the domain is restricted. Normally the domain is the whole of the real numbers and over that domain it must have at least one real zero.
The question has a simple answer only if the polynomial has rational coefficients. However, the question does not state that it has rational coefficients so it is not valid to assume that is the case. So, suppose the third root is p. Then the polynomial is (x + 3)*(x - 3 + 2i)*(x - p) = (x2 + 2xi - 9 + 6i)*(x - p) = x3 + (2i - p)x2 - (2pi - 6i + 9)x + (9p + 6pi)
The degree of a polynomial function is the highest power any single term is raised to. For example, (5a - 2b^2) is a second degree function because the "b^2" is raised to the second power and the "a" is only raised to the (implied) first power. For (24xy-xy^3 + x^2) it is a third degree polynomial because the highest power is the cube of -xy.
It is the Coefficient. It only refers to the given term that it is front. e.g. 2x^2 - 3x + 1 The '2' in front of 'x^2' only refers to 'x^2'. The '-3' in front of 'x' is the coefficient of '-3' The '1' is a constant.
5
No. Consider the polynomial: f(x) = x3 + 4x2 + 4x + 16 then f'(x) = 3x2 + 8x + 4 = (3x + 2)(x + 2) => x = -2/3, -2 are the zeros of f'(x) Using the second derivative: f''(x) = 6x + 8 it can be seen that: f''(-2) = -4 -> x = -2 is a maximum f''(-2/3) = +4 -> x = -2/3 is a minimum But plugging back into the original polynomial: f(-2) = 16 f(-2/3) = 14 22/27 Between the zeros of the first derivative, the slope of the polynomial is negative so that the polynomial is always decreasing in value, but as the polynomial is greater than zero at the zeros of the first derivative, it cannot become zero between them. That is it has no zeros between the zeros of its first derivative f(x) = x3 + 4x2 + 4x + 16 = (x + 4)(x2 + 4) has only 1 zero at x = -4.
You forgot to copy the polynomial. However, the Fundamental Theorem of Algebra states that every polynomial has at least one root, if complex roots are allowed. If a polynomial has only real coefficients, and it it of odd degree, it will also have at least one real solution.
Yes, easily. Even though the question did not ask what the polynomial was, only if I could find it, here is how you would find the polynomial: Since the coefficients are rational, the complex (or imaginary) roots must form a conjugate pair. That is to say, the two complex roots are + 3i and -3i. The third root is 7. So the polynomial, in factorised form, is (x - 3i)(x + 3i)(x - 7) = (x2 + 9)(x - 7) = x3 - 7x2 + 9x - 63