Ninth Grade Mathematics: Zeros & Roots of a Polynomial

In algebra, the zeros of a polynomial are the numerical values of the variable that cause the polynomial expression to evaluate to zero. Geometrically, they represent the exact coordinates where the graph of the polynomial intersects or touches the horizontal x-axis.

Defining Value, Zero, and Root

It is important to understand the precise terminology connecting polynomial values, zeros, and roots:

Terminology Connections:

1. Value of P(x) at x = a:
   The real number obtained by replacing every occurrence of x in P(x) with 'a',
   denoted by P(a).

2. Zero of a Polynomial P(x):
   A real number 'c' is called a zero of the polynomial P(x) if P(c) = 0.

3. Root of a Polynomial Equation:
   When a polynomial is set equal to zero to form an equation, P(x) = 0,
   the values of x that satisfy this equation are called the roots of the equation.

Thus, 'zeros' belong to polynomial expressions, while 'roots' belong to polynomial equations. A number c is a zero of P(x) if and only if c is a root of the equation P(x) = 0.

Example: Let P(x) = x^2 - 5*x + 6
Evaluate at x = 2:
P(2) = 2^2 - 5*(2) + 6 = 4 - 10 + 6 = 0.
Therefore, 2 is a zero of P(x), and x = 2 is a root of x^2 - 5*x + 6 = 0.

Geometric Interpretation on the Cartesian Plane

When we plot the graph of y = P(x) on the Cartesian coordinate plane, the zeros of the polynomial correspond exactly to the x-intercepts of the curve.

                     y
                     |             P(x) = x^2 - 4
                     |
                     |     *             *
                     |      *           *
                     |       *         *
    -----------------+--------*-------*--------+---- x
                    -2         *     *         2
                     |          *   *
                     |            *
                     |

In the diagram above for P(x) = x^2 - 4, the curve cuts the x-axis at x = -2 and x = 2. These two x-intercepts are precisely the zeros of the polynomial, since P(-2) = (-2)^2 - 4 = 0 and P(2) = 2^2 - 4 = 0.

Maximum Number of Zeros for a Polynomial

The Fundamental Theorem of Algebra guarantees a vital boundary regarding the number of zeros a polynomial can possess:

A non-zero polynomial P(x) of degree n can have at most n real zeros.

A linear polynomial (degree 1) has exactly one zero: a*x + b = 0 gives x = -b/a. A quadratic polynomial (degree 2) has at most two real zeros. A cubic polynomial (degree 3) has at most three real zeros. A non-zero constant polynomial P(x) = c (degree 0) has no zeros because c can never equal zero.

Polynomial Types and Maximum Zeros:
  Linear (Degree 1):      P(x) = 3*x - 6        ---> Zero at x = 2 (Exactly 1)
  Quadratic (Degree 2):   P(x) = x^2 - 9        ---> Zeros at x = 3, -3 (Max 2)
  Constant (Degree 0):    P(x) = 5              ---> No zeros (0 != 5)

Chapter Exercises

Exercise 1: Find all real zeros of each of the following polynomials: P(x) = 4x - 12, Q(x) = x^2 - 49, and R(x) = 2x^2 - 7*x + 3.

Exercise 2: Verify whether 3 and -1 are zeros of the cubic polynomial P(x) = x^3 - 3*x^2 - x + 3.

Exercise 3: Explain why a quadratic polynomial can have two distinct real zeros, one repeated real zero, or zero real zeros. Draw a simple ASCII sketch illustrating these three geometric possibilities.

Solutions and Explanations

Solution 1: To find the zeros of P(x) = 4x - 12, set P(x) = 0: 4x - 12 = 0, so 4x = 12, yielding x = 3. For Q(x) = x^2 - 49, set Q(x) = 0: x^2 - 49 = 0, which factors as (x - 7)(x + 7) = 0, giving zeros x = 7 and x = -7. For R(x) = 2x^2 - 7x + 3, factor the quadratic: 2x^2 - 6x - x + 3 = 2x(x - 3) - 1(x - 3) = (2x - 1)(x - 3) = 0. Setting each factor to zero gives 2x - 1 = 0 (so x = 1/2) and x - 3 = 0 (so x = 3). The zeros of R(x) are x = 1/2 and x = 3.

Solution 2: Substitute x = 3 into P(x): P(3) = 3^3 - 3(3^2) - 3 + 3 = 27 - 3(9) - 3 + 3 = 27 - 27 - 3 + 3 = 0. Because P(3) = 0, 3 is indeed a zero of P(x). Now substitute x = -1: P(-1) = (-1)^3 - 3((-1)^2) - (-1) + 3 = -1 - 3(1) + 1 + 3 = -1 - 3 + 1 + 3 = 0. Because P(-1) = 0, -1 is also a zero of P(x).

Solution 3: A quadratic polynomial graph is a parabola. Depending on its vertical position relative to the x-axis, the parabola can intersect the axis in three distinct ways:

    Two Real Zeros           One Real Zero           No Real Zeros
      (Cuts axis)            (Touches axis)          (Does not reach axis)
          \   /                   \   /                     \   /
           \ /                     \ /                       \ /
  ----------*---*-----     -----------*-------     -------------------
            |   |                     |

In the first case, the parabola dips below the axis and cuts it at two distinct points, producing two distinct real zeros. In the second case, the vertex of the parabola sits directly on the x-axis, touching it at a single point (a repeated zero of multiplicity 2). In the third case, the entire parabola lies strictly above the x-axis, never intersecting it, meaning the polynomial has zero real zeros.