By the Factor Theorem the expression x- a is a factor of a polynomialif and only if the value a is a zero of the related polynomial function. In other words, the expression x- a is a factor of a polynomial if and only if the graph of the related polynomial function has an x-intercept at ( a,0). We are given three points.
( - 3,0), (- 1,0), (1,0)We know that the graph of a polynomial has x-intercepts at these points. Therefore, the expressions x-( - 3), x-(- 1), and x-1 must be the factors of the polynomial. Let's multiply these three binomials.
The degree of a polynomial is the highest degree of its monomial.
x^3+3x^2-x-3
The degree of the above polynomial is 3. We obtained this polynomial by multiplying binomials that must be the factors of the described polynomial, so the expression x^3+3x^2-x-3 also must be the factor of that polynomial. Thus, the least possible degree of the described polynomial is 3.
