write an example of each of the following and represent themgeometrically on graph paper.
find the zeroes of the polynomials on the graph. futher factorize the polynomials to observe the zeroes there.
(a) constant polynomial
(b) linear polynomial
(c) quadratic polynomial
(d) cubic polynomial
(e) biquadratic polynomial
Answers
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.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:
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:
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:
x3
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.
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