`\color{green} ✍️` You know that a real number `k` is a zero of the polynomial `p(x)` if `p(k) = 0`.
But why are the zeroes of a polynomial so important ? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes.
Consider first a linear polynomial `ax + b, a ≠ 0`.
You have studied in Class IX that the graph of `y = ax + b` is a straight line.
For example, the graph of `y = 2x + 3` is a straight line passing through the points `(– 2, –1)` and `(2, 7)`.
You can see that the graph of `y = 2x + 3` intersects the `x` -axis mid-way between `x = –1` and `x = – 2`, that is, at the point ` ( -3/2 , 0 )` .
You also know that the zero of `2x + 3` is ` - 3/2` Thus, the zero of the polynomial `2x + 3` is the `x`-coordinate of the point
where the graph of `y = 2x + 3` intersects the `x-`axis.
In general, 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 )` .
Therefore, the linear polynomial `ax + b, a ≠ 0`, has exactly one zero, namely, the `x`-coordinate of the point where the graph of `y = ax + b` intersects the x-axis.
`color(red)(=>"Now, let us look for the geometrical meaning of a zero of a quadratic polynomial.")`
`color(blue)("Consider the quadratic polynomial")` `x^2 – 3x – 4`.
Let us see what the graph of `y = x^2 – 3x – 4` looks like.
Let us list a few values of `y = x^2 – 3x – 4` corresponding to a few values for `x` as given in Table `2.1.`
If we locate the points listed above on a graph paper and draw the graph, it will actually look like the one given in Fig. 2.2.
In fact, for any quadratic polynomial `ax^2 + bx + c, a ≠ 0`, the graph of the corresponding equation `y = ax^2 + bx + c` has one of the two shapes `color(blue)("either open upwards like or open downwards")` like depending on whether `a > 0` or `a < 0` . `color(green ("These curves are called parabolas.")`
You can see from Table 2.1 that `–1` and `4` are zeroes of the quadratic polynomial.
Also note from Fig. 2.2 that `–1` and `4` are the `x`-coordinates of the points where the graph of `y = x^2 – 3x – 4` intersects the `x`-axis.
Thus, `color(blue)("the zeroes of the quadratic polynomial")` `x^2 – 3x – 4` are x-coordinates of the points where the graph of
`y = x^2 – 3x – 4` intersects the `x`-axis
This fact is true for any quadratic polynomial, i.e., the zeroes of a quadratic polynomial `ax^2 + bx + c, a ≠ 0`, are precisely the `x`-coordinates of the points where the parabola representing `y = ax^2 + bx + c` intersects the x-axis.
`color(red)(=>ul"From our observation earlier about the shape of the graph of")` `y = ax^2 + bx + c`,
the following three cases can happen :
`color(red)("Case (i) :")` Here, the graph cuts `x`-axis at two distinct points `A` and `A′.`
The x-coordinates of `A` and `A′` are the two zeroes of the quadratic polynomial `ax^2 + bx + c` in this case (see Fig. 2.3).
`color(red)("Case (ii) :")` Here, the graph cuts the `x`-axis at exactly one point, i.e., at two coincident points.
So, the two points `A` and `A′` of `"Case (i)"` coincide here to become one point `A` (see Fig. 2.4).
The x-coordinate of `A` is the only zero for the quadratic polynomial` ax^2 + bx + c` in this case.
`color(red)("Case (iii) :")` Here, 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 (see Fig. 2.5).
So, the quadratic polynomial `ax^2 + bx + c` `color(blue)("has no zero in this case.")`
So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero.
`color(blue)("This also means that a polynomial of degree 2 has atmost two zeroes.")`
`color(red)(=>"Now, what do you expect the geometrical meaning of the zeroes of a cubic polynomial to be")` ? Let us find out.
`color(blue)("Consider the cubic polynomial " x^3 – 4x)`.
To see what the graph of `y = x^3 – 4x` looks like, let us list a few values of `y` corresponding to a few values for x as shown in Table 2.2.
Locating the points of the table on a graph paper and drawing the graph, we see that the graph of `y = x^3 – 4x ` actually looks like the one given in Fig. 2.6.
We see from the table above that `– 2, 0` and `2 ` are zeroes of the cubic polynomial `x^3 – 4x`.
Observe that `– 2, 0` and `2` are, in fact, the `x`-coordinates of the only points where the graph of `y = x^3 – 4x` intersects the x-axis.
Since the curve meets the x-axis in only these `3` points, their `x`-coordinates are the only zeroes of the polynomial.
Let us take a few more examples. Consider the cubic polynomials `x^3` and `x^3 – x^2`.
We draw the graphs of `y = x^3` and `y = x^3 – x^2` in Fig. 2.7 and Fig. 2.8 respectively.
Note that `0` is the only zero of the polynomial `x^3`.
Also, from Fig. 2.7, you can see that `0` is the x-coordinate of the only point where the graph of `y = x^3` intersects the `x`-axis.
Similarly, since` x^3 – x^2 = x^2 (x – 1)`, `" " 0` and `1` are the only zeroes of the polynomial `x^3 – x^2`.
Also, from Fig. 2.8, these values are the `x`-coordinates of the only points where the graph of `y = x^3 – x^2` intersects the x-axis.
From the examples above, we see that there are at most 3 zeroes for any cubic polynomial. In other words, any polynomial of degree `3` can have at most three zeroes.
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT) `
In general, given a polynomial `p(x)` of degree `n,` the graph of` y = p(x)` intersects the x-axis at atmost `n` points.
Therefore, a polynomial `p(x`) of degree `n` has at most `n` zeroes.
`\color{green} ✍️` You know that a real number `k` is a zero of the polynomial `p(x)` if `p(k) = 0`.
But why are the zeroes of a polynomial so important ? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes.
Consider first a linear polynomial `ax + b, a ≠ 0`.
You have studied in Class IX that the graph of `y = ax + b` is a straight line.
For example, the graph of `y = 2x + 3` is a straight line passing through the points `(– 2, –1)` and `(2, 7)`.
You can see that the graph of `y = 2x + 3` intersects the `x` -axis mid-way between `x = –1` and `x = – 2`, that is, at the point ` ( -3/2 , 0 )` .
You also know that the zero of `2x + 3` is ` - 3/2` Thus, the zero of the polynomial `2x + 3` is the `x`-coordinate of the point
where the graph of `y = 2x + 3` intersects the `x-`axis.
In general, 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 )` .
Therefore, the linear polynomial `ax + b, a ≠ 0`, has exactly one zero, namely, the `x`-coordinate of the point where the graph of `y = ax + b` intersects the x-axis.
`color(red)(=>"Now, let us look for the geometrical meaning of a zero of a quadratic polynomial.")`
`color(blue)("Consider the quadratic polynomial")` `x^2 – 3x – 4`.
Let us see what the graph of `y = x^2 – 3x – 4` looks like.
Let us list a few values of `y = x^2 – 3x – 4` corresponding to a few values for `x` as given in Table `2.1.`
If we locate the points listed above on a graph paper and draw the graph, it will actually look like the one given in Fig. 2.2.
In fact, for any quadratic polynomial `ax^2 + bx + c, a ≠ 0`, the graph of the corresponding equation `y = ax^2 + bx + c` has one of the two shapes `color(blue)("either open upwards like or open downwards")` like depending on whether `a > 0` or `a < 0` . `color(green ("These curves are called parabolas.")`
You can see from Table 2.1 that `–1` and `4` are zeroes of the quadratic polynomial.
Also note from Fig. 2.2 that `–1` and `4` are the `x`-coordinates of the points where the graph of `y = x^2 – 3x – 4` intersects the `x`-axis.
Thus, `color(blue)("the zeroes of the quadratic polynomial")` `x^2 – 3x – 4` are x-coordinates of the points where the graph of
`y = x^2 – 3x – 4` intersects the `x`-axis
This fact is true for any quadratic polynomial, i.e., the zeroes of a quadratic polynomial `ax^2 + bx + c, a ≠ 0`, are precisely the `x`-coordinates of the points where the parabola representing `y = ax^2 + bx + c` intersects the x-axis.
`color(red)(=>ul"From our observation earlier about the shape of the graph of")` `y = ax^2 + bx + c`,
the following three cases can happen :
`color(red)("Case (i) :")` Here, the graph cuts `x`-axis at two distinct points `A` and `A′.`
The x-coordinates of `A` and `A′` are the two zeroes of the quadratic polynomial `ax^2 + bx + c` in this case (see Fig. 2.3).
`color(red)("Case (ii) :")` Here, the graph cuts the `x`-axis at exactly one point, i.e., at two coincident points.
So, the two points `A` and `A′` of `"Case (i)"` coincide here to become one point `A` (see Fig. 2.4).
The x-coordinate of `A` is the only zero for the quadratic polynomial` ax^2 + bx + c` in this case.
`color(red)("Case (iii) :")` Here, 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 (see Fig. 2.5).
So, the quadratic polynomial `ax^2 + bx + c` `color(blue)("has no zero in this case.")`
So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero.
`color(blue)("This also means that a polynomial of degree 2 has atmost two zeroes.")`
`color(red)(=>"Now, what do you expect the geometrical meaning of the zeroes of a cubic polynomial to be")` ? Let us find out.
`color(blue)("Consider the cubic polynomial " x^3 – 4x)`.
To see what the graph of `y = x^3 – 4x` looks like, let us list a few values of `y` corresponding to a few values for x as shown in Table 2.2.
Locating the points of the table on a graph paper and drawing the graph, we see that the graph of `y = x^3 – 4x ` actually looks like the one given in Fig. 2.6.
We see from the table above that `– 2, 0` and `2 ` are zeroes of the cubic polynomial `x^3 – 4x`.
Observe that `– 2, 0` and `2` are, in fact, the `x`-coordinates of the only points where the graph of `y = x^3 – 4x` intersects the x-axis.
Since the curve meets the x-axis in only these `3` points, their `x`-coordinates are the only zeroes of the polynomial.
Let us take a few more examples. Consider the cubic polynomials `x^3` and `x^3 – x^2`.
We draw the graphs of `y = x^3` and `y = x^3 – x^2` in Fig. 2.7 and Fig. 2.8 respectively.
Note that `0` is the only zero of the polynomial `x^3`.
Also, from Fig. 2.7, you can see that `0` is the x-coordinate of the only point where the graph of `y = x^3` intersects the `x`-axis.
Similarly, since` x^3 – x^2 = x^2 (x – 1)`, `" " 0` and `1` are the only zeroes of the polynomial `x^3 – x^2`.
Also, from Fig. 2.8, these values are the `x`-coordinates of the only points where the graph of `y = x^3 – x^2` intersects the x-axis.
From the examples above, we see that there are at most 3 zeroes for any cubic polynomial. In other words, any polynomial of degree `3` can have at most three zeroes.
`\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT) `
In general, given a polynomial `p(x)` of degree `n,` the graph of` y = p(x)` intersects the x-axis at atmost `n` points.
Therefore, a polynomial `p(x`) of degree `n` has at most `n` zeroes.
