without solving, describe two ways to show that 2, -1, 3, and -2 are the roots of the polynomial equation x^2-2x^3-7x^2+8x+12=0.
plug in the values
graph the function
To show that 2, -1, 3, and -2 are the roots of the polynomial equation x^2 - 2x^3 - 7x^2 + 8x + 12 = 0 without solving it, we can use Vieta's formulas and observe the relationship between the polynomial's coefficients and its roots.
1. Using Vieta's Formulas:
Vieta's formulas state that if a polynomial equation of the form a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0 = 0 has roots r_1, r_2, ..., r_n, then the sum of the roots is given by -a_(n-1)/a_n and the product of the roots is given by (-1)^n * a_0/a_n.
In our case, the polynomial is x^2 - 2x^3 - 7x^2 + 8x + 12 = 0, which is a cubic polynomial (n = 3). According to Vieta's formulas, the sum of the roots should be -a_(n-1)/a_n = -8/1 = -8, and the product of the roots should be (-1)^n * a_0/a_n = (-1)^3 * 12/1 = -12.
Now, let's check if the roots 2, -1, 3, and -2 satisfy these conditions:
- The sum of the roots, 2 + (-1) + 3 + (-2) = 2 - 1 + 3 - 2 = 2, matches the value calculated using Vieta's formulas, which is -8. Hence, one way to show that these are the roots is by confirming the sum.
- The product of the roots, 2 * (-1) * 3 * (-2) = -12, also matches the value calculated using Vieta's formulas, which is -12. Therefore, another way to demonstrate that these are the roots is by verifying the product.
2. Observation of Coefficients:
By observing the coefficients of the polynomial equation x^2 - 2x^3 - 7x^2 + 8x + 12 = 0, we can infer some relationships between the roots and the equation:
- The term x^2 has a coefficient of -7, which indicates that the sum of the roots (2 + (-1) + 3 + (-2)) must contribute to this coefficient. As -7 is the sum of these values (-7 = 2 - 1 + 3 - 2), it suggests that 2, -1, 3, and -2 are indeed the roots.
- Additionally, the constant term 12 can be obtained by multiplying the roots together: 2 * (-1) * 3 * (-2) = -12.
Thus, by considering the sum and the product of the roots, as well as the relationship between the coefficients and the roots, we can assert that 2, -1, 3, and -2 are the roots of the polynomial equation x^2 - 2x^3 - 7x^2 + 8x + 12 = 0.
To show that 2, -1, 3, and -2 are the roots of the polynomial equation x^2-2x^3-7x^2+8x+12=0, we can use two methods:
1. Factor Theorem:
One way to show this is by applying the Factor Theorem. According to the theorem, if a number 'a' is a root of the polynomial equation, then (x - a) is a factor of the polynomial. Therefore, by substituting the given roots into the polynomial, we can check if they satisfy the equation and eliminate them as factors accordingly.
For example:
Substituting 2:
(2)^2 - 2(2)^3 - 7(2)^2 + 8(2) + 12 = 0,
4 - 16 - 28 + 16 + 12 = 0,
-12 - 12 + 12 = 0,
-12 = 0. (False)
Hence, 2 is not a root of the polynomial equation. Similarly, we can substitute -1, 3, and -2 to check if they satisfy the equation. If any of them give a value of 0 when plugged into the equation, they are the roots of the polynomial.
2. Vieta's Formulas:
Another method is to use Vieta's Formulas, which relate the coefficients of a polynomial to its roots. For a quadratic equation in the form ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.
In this case, the given polynomial equation is:
x^2-2x^3-7x^2+8x+12=0.
Comparing it to the general quadratic form, we have:
a = 1 (coefficient of x^2)
b = -2 (coefficient of x)
c = 12 (constant term)
According to Vieta's Formulas, the sum of the roots is equal to -b/a and the product of the roots is equal to c/a.
Sum of the roots:
-(-2)/1 = 2.
Product of the roots:
12/1 = 12.
Hence, the roots 2, -1, 3, and -2 satisfy the given polynomial equation x^2-2x^3-7x^2+8x+12=0, as they yield the correct sum and product values based on Vieta's Formulas.