Answer:
The factored form of the polynomial is [tex]y = (x-2)\cdot (x+1)[/tex].
Step-by-step explanation:
According to the graph, we have a second order polynomial with a vertical axis of symmetry. The standard form of the polynomial is:
[tex]y = a\cdot x^{2}+b\cdot x + c[/tex] (1)
Where:
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Dependent variable.
[tex]a[/tex], [tex]b[/tex], [tex]c[/tex] - Coefficients.
We can obtain the solution of the polynomial by knowing three distinct points. From graph we know that the curve pass through the following three points: [tex](x_{1},y_{1}) = (-1, 0)[/tex], [tex](x_{2},y_{2}) = (0, -2)[/tex], [tex](x_{3},y_{3}) = (2,0)[/tex]. Three linear equations are constructed:
[tex]a^{2}-b+c = 0[/tex] (2)
[tex]c = -2[/tex] (3)
[tex]4\cdot a + 2\cdot b + c = 0[/tex] (4)
The solution of this system is [tex]a = 1[/tex], [tex]b = -1[/tex] and [tex]c = -2[/tex]. Then, the polynomial in standard form is [tex]y = x^{2}-x-2[/tex]. By factorization, we find that factored form is:
[tex]y = (x-2)\cdot (x+1)[/tex] (5)
