Geometrical Meaning of the Zeros of a Polynomial: Explanation, Examples

Algebraic expressions having any number of finite terms with exponents as whole numbers are called polynomials. The zero of the polynomials is defined as any real value of \(x\), for which the value of the polynomial becomes zero. The zero of the polynomials is the \(x\)-coordinate of the point, where the graph intersects the \(X\)-axis. If a polynomial \(p(x)\) intersects the \(X\)-axis at \((k,\,0)\), then \(k\) is the zero polynomial. This article will discuss the Geometrical Meaning of the Zeros of a Polynomial for linear, quadratic and cubic polynomials in detail.

Geometric Meaning of Zeros of a Linear Polynomial

By displaying the \(x\) values on the \(X\)-axis and the associated polynomial values on the \(Y\)-axis, we can draw a graph of a polynomial \(p(x)\) on \(X – Y\) coordinate axes. The graph of a polynomial can be plotted as \(y = p\left( x \right)\) in a concise form.

A linear polynomial plotted as a straight line on X-Y coordinate axes

A straight line intersecting the \(X\)-axis at one point of zero of the polynomials always indicate a linear polynomial. A straight line parallel to the \(X\)-axis but does not cross it is represented by the equation \(y = b\), where \(b\) is a constant. A straight line with a zero coefficient of variable \(x\) cannot be considered a polynomial.

Therefore, a linear polynomial’s number of zeros is always \(1\). The linear polynomial \(p(x) = 7x + 3\) is plotted on the \(X – Y\) coordinate axes as follows:

At \(x = \frac{{ – 3}}{7},\,y = 0\) for the polynomial \(y = 7x + 3\). The polynomial zero is located at \(A\left( {\frac{{ – 3}}{7},\,0} \right)\).

The second point may be easily obtained by replacing \(x = 0\) for the value of \(y\), which is \(y = 3\). These are the coordinates of point \(B(0,\,3)\). To create a straight line, join the two points \(A\) and \(B\).

The point \(\left( {\frac{b}{a},\,0} \right)\) is the zero of a general linear polynomial \(y = ax + b\).

Learn About Zeros of a Polynomial

Geometric Meaning of Zeros of a Quadratic Polynomial

By using trial and error on simple values of \(x\) and drawing the graph for the quadratic polynomial \(y = {x^2} + 2x – 8\), we can quickly obtain the coordinates of the following points.

\(A( – 5,\,7),\,B(3,\,7),\,C( – 4,\,0),\,D(2,\,0),\,E( – 3,\, – 5),\,F(1,\, – 5),\,G( – 2,\, – 8),\,H(0,\, – 8)\), and \(I( – 1,\, – 9)\).

Put the values of \(x\) in the polynomial \({x^2} + 2x – 8\) for each coordinates to verify the values of \(y\). We’ll receive a roughly sketched curve if we connect the points. However, we show the actual plot for the quadratic polynomial in the following image.

The polynomial curve’s zeros are simply the two locations on the \(X\)-axis where the curve meets it, points \(C( – 4,\,0)\) and \(D(2,\,0)\).

Characteristics of Curve for a Quadratic Polynomial

Observe how the quadratic polynomial curve looks like a gradually narrower well, with a single point at the bottom where the value of \(y\) reaches its minimum and then rises as \(x\) increases. This is known as the quadratic polynomial minimum. The two arms of the curve increase equally on both sides of the minima point \(I\left( {1,\, – 9} \right)\). This is a parabolic curve.

The two arms cross the \(X\)-axis at the locations where the polynomial’s zeros are located and continue to rise endlessly. Surprisingly, the two rising arms of a curve on opposite sides of a line parallel to the \(X\)-axis and passing through the minima are exact mirror images of each other.

This is since any quadratic polynomial may be written as: 

\({(x + p)^2} \pm {q^2}\) where both \(p\) and \(q\) are real constants.

Indeed, the coordinates of the point where the polynomial’s value becomes lowest may be easily determined using this method. This is the usual form for obtaining quadratic polynomial minima (or maxima).

Now we can see that, although being parabolic, the curves for all quadratic polynomials are not the same. For some, the curve will resemble an inverted well with maxima rather than a minimum, and the number of zeros will be less than two. It might be \(1\) or even \(0\) in some cases.

Geometric Representation of a Quadratic Polynomial with Two Zeros but Having a Point of the Maximum Value of y: The polynomial \(y = p(x) =\, – {x^2} + 3x + 4\) has two zeros at locations \(A( – 1,\,0)\) and \(B(4,\,0)\), as well as one point \(C\left( {\frac{3}{2},\,\frac{{25}}{4}} \right)\) where the value of \(y\) reaches its maximum.

The value of \(y\) drops endlessly on both sides of this maximum.

The characteristic of such quadratic polynomials is that each polynomial must be a square of the sum of a linear polynomial.

For example, for \(y = {x^2} – 4x + 4\), the polynomial is actually,
\(y = p(x) = {(x – 2)^2}\).

Both zeroes of \(y\) have converged at \(x = 2\) i.e., \(B(2,\,0)\).

The curve crosses the \(X\)-axis only once and does not touch it geometrically. Similarly, in the second case,

\(y = q\left( x \right) = \, – {x^2} + 2x – 1 = \, – {\left( {x – 1} \right)^2}\)

As the curve approaches the \(X\)-axis, both zeros converge at a single point \(A(1,\,0)\).

Geometric Representation of a Quadratic Polynomial with No Zero

Two curves for quadratic polynomials, each with no real zero, are shown in the same figure.

The curves for the two forms of quadratic polynomials (with minima and with maxima) are shown in the same figure. The polynomials are as follows:

\(y = p(x) = {x^2} – 4x + 5\) and,
\(y = q\left( x \right) = \, – {x^2} + 2x – 2\)

For whatever value of \(x\), none of the curves reaches the \(X\)-axis. The minima are located above the \(X\)-axis, whereas the maximum is located below it. 

To summarise: The number of real zeros in quadratic polynomials can be \(0,\,1\) or \(2\), with a maximum of \(2\).

Geometric Meaning of Zeros of a Cubic Polynomial

The geometric representation of a cubic polynomial is shown below.

\(y = p\left( x \right) = {x^3} + 3{x^2} – 4x – 5\)

The cubic polynomial \(y = p(x) = {x^3} + 3{x^2} – 4x – 5\) has a maximum at point \(A\) and a minimum at point \(B\) on the curve plot.

On the left, as \(x\) increases, the value of \(y\) rises crosses the \(X\)-axis, and begins to decline after reaching the maximum. This decline continues, and the curve crosses the \(X\)-axis for the second time on its way to \(B\), where it reaches its minima.

With rising \(x\), the value of \(y\) continues to rise, crossing the \(X\)-axis for the third time and growing forever.

This cubic polynomial curve features three zeros, which are the locations where the curve intersects the \(X\)-axis. The maximum number of zeros that a cubic polynomial may have is three. The following shows the second case of two real zeros for a cubic polynomial. The third zero will be imaginary.

The cubic equation \({x^3} + 4{x^2}\) intersects the \(X\)-axis at the point \((0,\,0)\) and \(( – 4,\,0)\). Zero of a polynomial \({x^3} + 4{x^2}\) are the \(x\)-coordinates of the point where the graph cuts the \(X\)-axis. Zeroes of the cubic polynomial are \(-4\) and \(0\).

The following shows the third case of one real zero of a cubic polynomial.

The cubic polynomial \(y = {x^3} + 2{x^2} – 6x + 5\) has only one zero.

Note: In this case, there are two imaginary roots and only one real-valued root. In general, the number of zeros of an \(n\)-degree polynomial is at most \(n\).

Relationship between Zeroes and Coefficients of a Polynomial

Solved Examples on Geometrical Meaning of the Zeros of a Polynomial

Q.1. The graph of \(y = p(x)\) for the polynomial \(p(x)\) is drawn below. Find the number of zeroes of the polynomial.

Ans: The graph of the given polynomial \(p(x)\) is a straight line parallel to the \(X\)-axis, i.e., \(y = p\left( x \right) = {\rm{constant}}{\rm{.}}\) It will never intersect the \(X\)-axis, and so the number of zeros of the given polynomial is zero.

Q.2. The graph of \(y = p(x)\) for the polynomial \(p(x)\) is drawn below. Find the number of zeroes of the polynomial.

Answer: The graph of the given polynomial \(p(x)\) is a cubic polynomial that intersects the \(X\)-axis only at once. So, the number of zeros of the given polynomial is one.

Q.3. Find the roots of the polynomial \(p(x) = {x^2} + 2x – 15\)
Ans: Given, a polynomial \(p(x) = {x^2} + 2x – 15\)
Put \(p(x) = 0\),
\({x^2} + 2x – 15 = 0\)
\( \Rightarrow {x^2} + 5x – 3x – 15 = 0\)
\( \Rightarrow x(x + 5) – 3(x + 5) = 0\)
\( \Rightarrow (x – 3)(x + 5) = 0\)
Hence, the roots of the given polynomial are\(x = 3\) and \(x = \,- 5\). 

Q.4. Find the roots of the quadratic equation, \({x^2} + 6x – 14 = 0\)
Ans: Given, \({x^2} + 6x – 14 = 0\)
To find the roots of the quadratic equation \(a{x^2} + bx + c = 0\), we will use
\(x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\)
Therefore, the roots of the given equation are
\(x = \frac{{ – 6 \pm \sqrt {{6^2} – 4 \times 1 \times \left( { – 14} \right)} }}{{2 \times 1}}\)
\(x = \frac{{ – 6 \pm \sqrt {36 + 56} }}{2}\)
\(x = \frac{{ – 6 \pm \sqrt {92} }}{2}\)
\(x = \, – 3 \pm \sqrt {23} \)
Hence, the roots of the quadratic equation are \( – 3 + \sqrt {23} \) and \( – 3 – \sqrt {23} .\)

Q.5. Identify the point where the graph of the equation \({x^2} – 2x – 8 = 0\) intersects \(X\)-axis.
Ans: Given, a quadratic equation \({x^2} – 2x – 8 = 0\)
Now, factorize the given equation,
\({x^2} – 2x – 8 = 0\)
\( \Rightarrow {x^2} – 4x + 2x – 8 = 0\)
\( \Rightarrow x(x – 4) + 2(x – 4) = 0\)
\( \Rightarrow (x – 4)(x + 2) = 0\)
\(x = 4,\,x = \, – 2\)
Hence, the equation will intersect the graph on \(X\)-axis at points \((4,\,0)\) and \(( – 2,\,0).\)

Summary

So we discussed the geometrical meaning of zeros of a linear polynomial, geometrical meaning of zeros of a quadratic polynomial and geometrical meaning of zeros of a cubic polynomial with the solved examples and frequently asked questions. Geometrically, a linear polynomial is a straight line that intersects the X axis and Y axis at maximum 1 point each; a quadratic polynomial is an upward or a downward curved parabola that intersects the X axis at maximum 2 points and Y axis at maximum 1 point. The cubic polynomial is a curve that starts asymptotically to the negative Y axis, reaches a maximum point and curves downwards, reaches a minimum point and curves upwards, and travels asymptotically to the positive Y axis. It cuts the X axis at a maximum of 3 points and Y axis at a maximum of 1 point.

FAQs on Geometrical Meaning of the Zeros of a Polynomial

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