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Polynomials

Geometrical Representation of Zeroes of a Polynomial

In the study of polynomials, you are aware that a real number ‘k’ is a zero of the polynomial p(x) if p(k) = 0. Remember, zero of a polynomial is different from a zero polynomial. We have seen the Remainder theorem use the concept of zeroes of a polynomial too. In order to understand their importance, we will look at the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes.

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Linear Polynomial

Let’s look at a linear polynomial ax + b, where a ≠ 0. You have already studied that the graph of y = ax + b is a straight line. Let’s look at the graph of y = 2x + 3.

x -2 2
y = 2x + 3 -1 7

The straight line y = 2x + 3 will pass through the points (- 2, – 1) and (2, 7). Here is how the graph looks like:

zeros

Fig. 1

From the Fig.1 above, you can see that the graph of y = 2x + 3 intersects the x-axis at the point (- 3/2, 0). Now, the zero of (2x + 3) is (- 3/2). Therefore, the zero of the linear polynomial (2x + 3) is the x-coordinate of the point where the graph of y = 2x + 3 intersects the x-axis. Hence, we can say,

For a linear polynomial ax + b, a ≠ 0, the graph of y = ax + b is a straight line which intersects the x-axis at exactly one point, namely, (- b/a, 0). Also, this linear polynomial has only one zero which is the x-coordinate of the point where the graph of y = ax + b intersects the x-axis.

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Quadratic Polynomial

Let’s look at a quadratic polynomial, x2 – 3x – 4. To look at the graph of y = x2 – 3x – 4, let’s list some values:

x – 2 – 1 0 1 2 3 4 5
y = x2 – 3x – 4 6 0 – 4 – 6 – 6 – 4 0 6

The graph of y = x2 – 3x – 4 will pass through (- 2, 6), (- 1, 0), (0, – 4), (1, – 6), (2, – 6), (3, – 4), (4, 0) and (5, 6). Here is how the graph looks:

zero polynomial

Fig. 2

For that matter, for any quadratic polynomial y = ax2 + bx + c, a ≠ 0, the graph of y = ax2 + bx + c has either one of these two shapes:

  • If a > 0, then it is open upwards like the one shown in the graph above
  • If a < 0, then it is open downwards.

These curves are parabolas. A quick look at the table above shows that (-1) and (4) are zeroes of the quadratic polynomial. From the Fig. 2 above, you can see that (-1) and (4) are the x-coordinates of the points where the graph of y = x2 – 3x – 4 intersects the x-axis. Therefore, we can say,

The zeroes of a quadratic polynomial ax2 + bx + c, a ≠ 0, are precisely the x-coordinates of the points where the parabola representing y = ax2 + bx + c intersects the x-axis.

As far as the shape of the graph is concerned, the following three cases are possible:

Case (i)

The graph cuts x-axis at two distinct points A and A′, where the x-coordinates of A and A′ are the two zeroes of the quadratic polynomial ax2 + bx + c, as shown below:

zero polynomial

Fig. 3

Case (ii)

The graph intersects the x-axis at only one point, or at two coincident points. Also, the x-coordinate of A is the only zero for the quadratic polynomial ax2 + bx + c, as shown below:

zero polynomial

Fig. 4

Case (iii)

The graph is either

  • Completely above the x-axis or
  • Completely below the x-axis.

So, it does not cut the x-axis at any point. Hence, the quadratic polynomial ax2 + bx + c has no zero, as shown below:

zero polynomial

Fig. 5

To summarize, a quadratic polynomial can have either:

  • Two distinct zeroes (as shown in Case i)
  • Two equal zeroes (or one zero as shown in Case ii)
  • No zero (as shown in Case iii)

It can also be summarized by saying that a polynomial of degree 2 has a maximum of 2 zeroes.

Cubic Polynomial

Let’s look at a cubic polynomial, x3 – 4x. Next, let’s list a few values to plot the graph of y = x3 – 4x.

x – 2 – 1 0 1 2
y = x3 – 4x 0 3 0 – 3 0

The graph of y = x3 – 4x will pass through (- 2, 0), (- 1, 3), (0, 0), (1, – 3), and (2, 0). Here is how the graph looks like:

zero polynomial

Fig. 6

From the table above, we can see that 2, 0 and – 2 are the zeroes of the cubic polynomial x3 – 4x. You can also observe that the graph of y = x3 – 4x intersects the x-axis at 2, 0 and – 2. Let’s take a quick look at some examples:

Let’s plot the graph of the following two cubic polynomials:

  1. x3
  2. x3 – x2

The graphs of y = x3 and y = x3 – x2 look as follows:

zero polynomial

Fig. 7

From the first graph, you can observe that 0 is the only zero of the polynomial x3, since the graph of y = x3 intersects the x-axis only at 0. Similarly, the polynomial x3 – x2 = x2(x – 1) has two zeroes, 0 and 1. From the second diagram, you can see that the graph of y = x3 – x2 intersects the x-axis at 0 and 1.

Hence, we can conclude that there is a maximum of three zeroes for any cubic polynomial. Or, any polynomial with degree 3 can have maximum 3 zeroes. In general,

Given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at a maximum of n points. Therefore, a polynomial p(x) of degree n has a maximum of n zeroes.

Solved Examples for You

Question 1: The graphs of y = p(x) are given in the figure below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

zero polynomial

Fig. 8

Answer : The number of zeroes in each of the graphs above are:

  • 0, since the line is not intersecting the x-axis.
  • 1, since the line intersects the x-axis once.
  • 2, since the line intersects the x-axis twice.
  • 2, since the line intersects the x-axis twice.
  • 4, since the line intersects the x-axis four times.
  • 3, since the line intersects the x-axis thrice.

Question 2:  What is meant by zeros of the function?

Answer: The zero of a function refers to any replacement for the variable that shall result in an answer of zero. Graphically, the real zero of a function is where the function’s graph crosses the x‐axis. So, the real zero of a function is the x‐intercept(s) of the function’s graph.

Question 3: How can one find zeroes in a function?

Answer: In order to find the zero of a function, one must find the point (a, 0) where the intersection of the function’s graph and the y-intercept takes place. In order to find the value of a from the point (a, 0), one must set the function equal to zero. Afterwards, one must try to solve for x.

Question 4: How many zeroes exist in a linear polynomial?

Answer: A linear polynomial consists of only 1 zero. In contrast, a quadratic polynomial consists of 2 zeroes while a cubic polynomial consists of 3 zeroes.

Question 5: Explain the zeroes in an equation?

Answer: The zeros of a quadratic equation refer to the points where the crossing of the graph of the quadratic equation takes place to the x-axis.

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