what does the polynomial represent
a.-x^3+2x^2-3x+2
b.x^2+x-2
c.-x^2-x+2
d.-x^2+x-2
a. A third degree polynomial with decreasing coefficients.
b. A second degree polynomial that can be factored as (x-1)(x+2).
c. A second degree polynomial that can be factored as -(x-1)(x-2).
d. A second degree polynomial that can be factored as -(x+1)(x-2).
The polynomials listed represent mathematical functions in the form of ax^n + bx^(n-1) + ... + k, where a, b, ... , k are coefficients and n is the highest power of x in the polynomial.
a. -x^3 + 2x^2 - 3x + 2 represents a cubic function, with x raised to the power of 3 as the highest power.
b. x^2 + x - 2 represents a quadratic function, with x raised to the power of 2 as the highest power.
c. -x^2 - x + 2 also represents a quadratic function, with x raised to the power of 2 as the highest power.
d. -x^2 + x - 2 also represents a quadratic function, with x raised to the power of 2 as the highest power.
The given options are all polynomials, which are algebraic expressions consisting of variables raised to non-negative integer powers (in this case, x) and constant coefficients. Polynomials are used to represent various mathematical quantities, such as functions, equations, or mathematical models.
To determine what each polynomial represents, we can analyze their structure and properties. Let's go through each option:
a. -x^3 + 2x^2 - 3x + 2
This cubic polynomial represents a function that involves powers of x up to the third degree. By analyzing its coefficients and terms, we can deduce various properties of this polynomial, such as its roots, turning points, and behavior at different x-values.
b. x^2 + x - 2
This quadratic polynomial represents a function that involves powers of x up to the second degree. It is a standard quadratic equation, which can be factored or solved using the quadratic formula to find its roots. The graph of this polynomial will be a parabola.
c. -x^2 - x + 2
This quadratic polynomial, similar to option b, represents a function involving powers of x up to the second degree. By comparing its coefficients and terms, we can analyze its properties, such as its vertex, axis of symmetry, and whether it opens upwards or downwards.
d. -x^2 + x - 2
This quadratic polynomial, like options b and c, represents a function involving powers of x up to the second degree. By examining its coefficients and terms, we can deduce its properties, such as its vertex, axis of symmetry, and concavity.
To better understand what each polynomial represents, it is helpful to graph them or perform further analysis based on their coefficients and terms. This will provide more insights into their behavior, roots, and other characteristics.