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Yes, the leading coefficient of a polynomial function can be a fraction. A polynomial is defined as a sum of terms, each consisting of a coefficient (which can be any real number, including fractions) multiplied by a variable raised to a non-negative integer power. Thus, the leading coefficient, which is the coefficient of the term with the highest degree, can indeed be a fraction.
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What is type a polynomial with integer coefficients and a leading coefficient of 1 in the box below?
A polynomial with integer coefficients and a leading coefficient of 1 is called a monic polynomial. An example of such a polynomial is ( f(x) = x^3 - 4x^2 + 6x - 2 ). In this polynomial, all coefficients are integers, and the leading term ( x^3 ) has a coefficient of 1.
What is a polynomial function f of least degree that has rational coefficient a leading coefficient of 1 and the given zeros -7 -4?
A polynomial function of least degree with rational coefficients and a leading coefficient of 1 that has the zeros -7 and -4 can be constructed using the fact that if ( r ) is a zero, then ( (x - r) ) is a factor. Therefore, the polynomial can be expressed as ( f(x) = (x + 7)(x + 4) ). Expanding this, we get ( f(x) = x^2 + 11x + 28 ). Thus, the polynomial function is ( f(x) = x^2 + 11x + 28 ).
Give the degree and the leading coefficient of the polynomial 9x-45x- squared -4x to the third power?
The polynomial can be rewritten as (-4x^3 - 45x^2 + 9x). The degree of the polynomial is 3, which is determined by the highest exponent of (x). The leading coefficient, which is the coefficient of the term with the highest degree, is (-4).
If the degree of a polynomial function is even and the leading coefficient is negative then what will happen to the ends?
If a polynomial function has an even degree and a negative leading coefficient, the ends of the graph will both point downward. This means that as the input values approach positive or negative infinity, the output values will also approach negative infinity. In summary, the graph will have a "U" shape that opens downwards.
What is the leading term in a polynomial?
The leading term in a polynomial is the term with the highest degree, which determines the polynomial's end behavior and its classification (e.g., linear, quadratic, cubic). It is typically expressed in the form ( ax^n ), where ( a ) is a non-zero coefficient and ( n ) is a non-negative integer. The leading term is crucial for understanding the polynomial's growth as the input values become very large or very small.
What is the leading coefficient in a polynomial?
It is the number (coefficient) that belongs to the variable of the highest degree in a polynomial.
The rational roots of a polynomial function F(x) can be written in the form where p is a factor of the constant term of the polynomial and q is a factor of the leading coefficient.?
TRue
What is type a polynomial with integer coefficients and a leading coefficient of 1 in the box below?
A polynomial with integer coefficients and a leading coefficient of 1 is called a monic polynomial. An example of such a polynomial is ( f(x) = x^3 - 4x^2 + 6x - 2 ). In this polynomial, all coefficients are integers, and the leading term ( x^3 ) has a coefficient of 1.
What is a polynomial function f of least degree that has rational coefficient a leading coefficient of 1 and the given zeros -7 -4?
A polynomial function of least degree with rational coefficients and a leading coefficient of 1 that has the zeros -7 and -4 can be constructed using the fact that if ( r ) is a zero, then ( (x - r) ) is a factor. Therefore, the polynomial can be expressed as ( f(x) = (x + 7)(x + 4) ). Expanding this, we get ( f(x) = x^2 + 11x + 28 ). Thus, the polynomial function is ( f(x) = x^2 + 11x + 28 ).
Give the degree and the leading coefficient of the polynomial 9x-45x- squared -4x to the third power?
The polynomial can be rewritten as (-4x^3 - 45x^2 + 9x). The degree of the polynomial is 3, which is determined by the highest exponent of (x). The leading coefficient, which is the coefficient of the term with the highest degree, is (-4).
Where p is a factor of the leading coefficient of the polynomial and q is a factor of the constant term.?
Anywhere. Provided it is not zero, and number p can be the leading coefficient of a polynomial. And any number q can be the constant term.
What is the number in front of the term with the highest degree in a polynomial?
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.
What is a possible leading coefficient and degree for a polynomial starting in quadrant 3 and ending in quadrant 4?
Leading coefficient: Negative. Order: Any even integer.
What is the leading term in a polynomial?
The leading term in a polynomial is the term with the highest degree, which determines the polynomial's end behavior and its classification (e.g., linear, quadratic, cubic). It is typically expressed in the form ( ax^n ), where ( a ) is a non-zero coefficient and ( n ) is a non-negative integer. The leading term is crucial for understanding the polynomial's growth as the input values become very large or very small.
What is the rational zero theorem?
If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form ± p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
What is 2X squared minus 7X -4?
The expression (2X^2 - 7X - 4) is a quadratic polynomial in the variable (X). It represents a parabolic function when graphed, with a leading coefficient of 2 indicating that the parabola opens upwards. The roots of this polynomial can be found using the quadratic formula, and it can also be factored if possible.
Which polynomial has rational coefficients a leading leading coefficient of 1 and the zeros at 2-3i and 4?
There cannot be such a polynomial. If a polynomial has rational coefficients, then any complex roots must come in conjugate pairs. In this case the conjugate for 2-3i is not a root. Consequently, either (a) the function is not a polynomial, or (b) it does not have rational coefficients, or (c) 2 - 3i is not a root (nor any other complex number), or (d) there are other roots that have not been mentioned. In the last case, the polynomial could have any number of additional (unlisted) roots and is therefore indeterminate.
