The polynomial x^2 + 2x - 3 has zeroes 1 and -3, and the relation between the coefficients and zeroes is verified.
Polynomial with zeroes 1 and -3:
The zeroes of a polynomial are the values of x that make the equation equal to 0. Therefore, we need to construct a polynomial where x = 1 and x = -3 make the equation equal to 0.
Step 1: Construct the polynomial
We know that a polynomial with zeroes at 1 and -3 can be factored as follows:
(x - 1)(x + 3) = 0
Expanding this expression gives us the quadratic polynomial:
x^2 + 2x - 3 = 0
Step 2: Verify the relation between coefficients and zeroes
There is a relationship between the coefficients of a polynomial and its zeroes. For a quadratic polynomial of the form:
ax^2 + bx + c = 0
The sum of the zeroes is equal to -b/a and the product of the zeroes is equal to c/a.
In our case:
a = 1
b = 2
c = -3
Therefore, according to the relationship:
Sum of zeroes = -b/a = -2/1 = -2
Product of zeroes = c/a = -3/1 = -3
Checking with our zeroes 1 and -3:
Sum of zeroes = 1 + (-3) = -2 (correct)
Product of zeroes = 1 * (-3) = -3 (correct)
Therefore, the polynomial x^2 + 2x - 3 has zeroes 1 and -3, and the relation between the coefficients and zeroes is verified.
Complete question:
Forma de polynomial whose zeroes are 1 & -3.
Verify the relation between the coefficients & the zeroes of
the polynomial