01 · Explore
Introduction to Polynomials
A polynomial is an algebraic expression consisting of variables and coefficients. The nature of a polynomial is primarily determined by its degree.
In a polynomial p(x), the highest power of the variable x is called the degree of the polynomial. For instance, 4x + 2 is a polynomial of degree 1, while 2y² - 3y + 4 is a polynomial of degree 2. It is important to note that expressions involving variables in the denominator or under a square root, such as 1/(x-1) or √x + 2, are not considered polynomials.
Polynomials are classified based on their degree. A polynomial of degree 1 is a linear polynomial (e.g., 2x - 3). A polynomial of degree 2 is a quadratic polynomial, derived from the word 'quadrate' meaning square (e.g., x² + 3x - 2). A polynomial of degree 3 is known as a cubic polynomial (e.g., 2 - x³).
The general form of a quadratic polynomial in x is ax² + bx + c, where a, b, and c are real numbers and a ≠ 0. Similarly, the general form of a cubic polynomial is ax³ + bx² + cx + d, where a ≠ 0.
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Question
What is the degree of the polynomial p(u) = 7u⁶ - 3u⁴ + 4u² + u - 8?
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02 · Explore
Zeroes of a Polynomial
The value of a polynomial at a specific point helps us identify its 'zeroes', which are the inputs that make the entire expression equal to zero.
If p(x) is a polynomial and k is any real number, the value obtained by replacing x with k in p(x) is denoted as p(k). For example, if p(x) = x² - 3x - 4, then p(2) = 2² - 3(2) - 4 = 4 - 6 - 4 = -6.
A real number k is said to be a zero of a polynomial p(x) if p(k) = 0. In the case of p(x) = x² - 3x - 4, we find that p(-1) = (-1)² - 3(-1) - 4 = 1 + 3 - 4 = 0. Similarly, p(4) = 4² - 3(4) - 4 = 16 - 12 - 4 = 0. Therefore, -1 and 4 are the zeroes of this quadratic polynomial.
For a linear polynomial ax + b (where a ≠ 0), finding the zero is straightforward: we set ax + b = 0, which gives x = -b/a. Thus, a linear polynomial has exactly one zero, which is -(Constant term) / (Coefficient of x).
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Find the zero of the linear polynomial p(x) = 2x + 3.
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Geometrical Meaning of Zeroes
The zeroes of a polynomial have a distinct visual representation when the polynomial is plotted on a coordinate plane.
The graph of a linear polynomial y = ax + b is always a straight line. The zero of this polynomial is the x-coordinate of the point where the line intersects the x-axis. For y = 2x + 3, the line crosses the x-axis at (-3/2, 0).
For a quadratic polynomial y = ax² + bx + c, the graph is a curve called a parabola. If a > 0, the parabola opens upwards (like a U); if a < 0, it opens downwards (like an inverted U).
The zeroes of a quadratic polynomial are the x-coordinates of the points where the parabola intersects the x-axis. Depending on the polynomial, the graph may intersect the x-axis at two distinct points, touch it at exactly one point (two coincident points), or not intersect it at all.
Visualizing Zeroes on a Graph
- 1
Plot y = p(x)
Create a table of values for x and y, then plot these points on a Cartesian plane.
- 2
Identify Intersections
Look for points where the graph crosses or touches the x-axis. Both have y = 0.
- 3
Extract x-coordinates
The x-values at these intersection points are the zeroes of the polynomial.
The number of times the graph intersects the x-axis indicates the number of zeroes the polynomial has.
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Question
If a parabola does not touch or cross the x-axis at any point, how many real zeroes does the quadratic polynomial have?
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04 · Explore
Zeroes of Cubic Polynomials
Cubic polynomials follow a similar geometrical pattern but can have more complex curves.
A cubic polynomial p(x) = ax³ + bx² + cx + d can have at most three zeroes. Geometrically, this means the graph of a cubic polynomial can intersect the x-axis at a maximum of three points.
For example, the graph of y = x³ - 4x intersects the x-axis at x = -2, x = 0, and x = 2. These three values are the zeroes of the polynomial. In contrast, the polynomial y = x³ has only one zero at x = 0, where the graph meets the x-axis.
In general, a polynomial of degree n can have at most n zeroes. This is a fundamental property: the number of real zeroes is always less than or equal to the degree of the polynomial.
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What is the maximum number of zeroes a polynomial of degree 4 can have?
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