Comprehensive Study Notes on Polynomials: Degrees, Zeroes, and Geometrical Interpretations

Introduction and Definitions of Polynomials

• Definition of Degree: In a polynomial $p(x)$, the highest power of $x$ in $p(x)$ is called the degree of the polynomial.

• Variable Examples:

  • $4x + 2$ is a polynomial in variable $x$ of degree $1$.

  • $2y^{2} - 3y + 4$ is a polynomial in variable $y$ of degree $2$.

  • $5x^{3} - 4x^{2} + x - \sqrt{2}$ is a polynomial in variable $x$ of degree $3$.

  • $7u^{6} - \frac{3}{4}u^{4} + 4u^{2} + u - 8$ is a polynomial in variable $u$ of degree $6$.

• Non-Polynomials: Expressions such as $\frac{1}{x - 1}$, $\sqrt{x} + 2$, and $\frac{1}{x^{2} + 2x + 3}$ are not considered polynomials.

Classification of Polynomials by Degree

• Linear Polynomial: A polynomial of degree $1$.

  • Examples: $2x - 3$, $\sqrt{3}x + 5$, $y + \sqrt{2}$, $x - \frac{2}{11}$, $\frac{2}{3}u + 1$, and $z + \frac{3}{4}$.

  • Counter-examples: $2x + 5 - x^{2}$ and $x^{3} + 1$ are not linear polynomials.

• Quadratic Polynomial: A polynomial of degree $2$. The name 'quadratic' comes from the word 'quadrate', meaning 'square'.

  • General Form: $ax^{2} + bx + c$, where $a, b, c$ are real numbers and $a \neq 0$.

  • Examples: $2x^{2} + 3x - \frac{2}{5}$, $y^{2} - 2$, $2 - x^{2} + \sqrt{3}x$, $\frac{u}{3} - 2u^{2} + 5$, $\sqrt{5}v^{2} - \frac{2}{3}v$, and $4z^{2} + \frac{1}{7}$.

• Cubic Polynomial: A polynomial of degree $3$.

  • General Form: $ax^{3} + bx^{2} + cx + d$, where $a, b, c, d$ are real numbers and $a \neq 0$.

  • Examples: $2 - x^{3}$, $x^{3}$, $\sqrt{2}x^{3}$, $3 - x^{2} + x^{3}$, and $3x^{3} - 2x^{2} + x - 1$.

Value and Zeroes of a Polynomial

• Value at a Point: If $p(x)$ is a polynomial in $x$, and $k$ is any real number, the value obtained by replacing $x$ with $k$ in $p(x)$ is denoted as $p(k)$.

  • Calculation Example: For $p(x) = x^{2} - 3x - 4$, at $x = 2$: $p(2) = 2^{2} - (3 \times 2) - 4 = 4 - 6 - 4 = -6$.

  • Calculation Example: At $x = 0$: $p(0) = 0^{2} - (3 \times 0) - 4 = -4$.

• Zero of a Polynomial: A real number $k$ is said to be a zero of a polynomial $p(x)$ if $p(k) = 0$.

  • Example Case: For $p(x) = x^{2} - 3x - 4$, we find $p(-1) = (-1)^{2} - 3(-1) - 4 = 1 + 3 - 4 = 0$ and $p(4) = 4^{2} - 3(4) - 4 = 16 - 12 - 4 = 0$. Thus $-1$ and $4$ are the zeroes of this quadratic polynomial.

• Zero of a Linear Polynomial: For $p(x) = ax + b$, let $k$ be the zero. Then $p(k) = ak + b = 0$, which leads to $k = \frac{-b}{a}$.

  • General Relationship: The zero of the linear polynomial $ax + b$ is $\frac{-(\text{Constant term})}{\text{Coefficient of } x}$.

Geometrical Meaning of Zeroes

• Linear Polynomial Geometrically: The graph of $y = ax + b$ is always a straight line. The zero of the polynomial is the $x$-coordinate of the point where this line intersects the $x$-axis. For example, $y = 2x + 3$ passes through $( -2, -1 )$ and $( 2, 7 )$ and intersects the $x$-axis at $( \frac{-3}{2}, 0 )$.

• Quadratic Polynomial Geometrically (Parabolas): The graph of $y = ax^{2} + bx + c$ is a curve called a parabola.

  • Orientation: If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards.

  • Three Geometrical Cases for Zeroes:

    • Case (i): The graph cuts the $x$-axis at two distinct points $A$ and $A'$. The $x$-coordinates are the two distinct zeroes.

    • Case (ii): The graph touches the $x$-axis at exactly one point ($A$ and $A'$ coincide). There is exactly one zero (or two equal zeroes).

    • Case (iii): The graph is completely above or below the $x$-axis and does not cut it. There are no real zeroes.

• At Most Principle: A quadratic polynomial (degree $2$) has at most $2$ zeroes. A cubic polynomial (degree $3$) has at most $3$ zeroes. Generally, a polynomial of degree $n$ has at most $n$ zeroes.

• Cubic Illustration: For $y = x^{3} - 4x$, the zeroes are $-2$, $0$, and $2$. These are the points where the graph intersects the $x$-axis.

Relationship between Zeroes and Coefficients

• Quadratic Relationship:

  • If $\alpha$ and $\beta$ are zeroes of $p(x) = ax^{2} + bx + c$, then:

  • $\text{Sum of zeroes} = \alpha + \beta = \frac{-b}{a} = \frac{-(\text{Coefficient of } x)}{\text{Coefficient of } x^{2}}$

  • $\text{Product of zeroes} = \alpha\beta = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^{2}}$

• Cubic Relationship:

  • If $\alpha$, $\beta$, and $\gamma$ are zeroes of $p(x) = ax^{3} + bx^{2} + cx + d$, then:

  • $\text{Sum of zeroes} = \alpha + \beta + \gamma = \frac{-b}{a}$

  • $\text{Sum of products taken two at a time} = \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$

  • $\text{Product of zeroes} = \alpha\beta\gamma = \frac{-d}{a}$

Worked Examples and Problem Solutions

• Example 1: Finding zeroes from graphs. The number of zeroes is determined by the number of times the graph intersects the $x$-axis. (e.g., Graph i = 1 zero, Graph ii = 2 zeroes, Graph iii = 3 zeroes).

• Example 2: Assertion and Reason.

  • Assertion (A): If the graph intersects the $x$-axis at only one point, it cannot be quadratic. (False, because a quadratic can have two equal zeroes touching at one point).

  • Reason (R): A polynomial of degree $n$ has at most $n$ real zeroes. (True).

• Example 3: Verification of Quadratic Zeroes. For $x^{2} + 7x + 10$, splitting the middle term gives $(x+2)(x+5)$. Zeroes are $-2$ and $-5$.

  • $\text{Sum} = -2 + (-5) = -7 = \frac{-7}{1}$.

  • $\text{Product} = (-2) \times (-5) = 10 = \frac{10}{1}$.

• Example 4: Polynomial difference of squares. For $x^{2} - 3$, zeroes are $\sqrt{3}$ and $-\sqrt{3}$.

  • $\text{Sum} = \sqrt{3} - \sqrt{3} = 0 = \frac{0}{1}$.

  • $\text{Product} = (\sqrt{3}) \times (-\sqrt{3}) = -3 = \frac{-3}{1}$.

• Example 5: Matching Columns.

  • If $a+b+c=0$ for $ax^{2}+bx+c$, then $1$ is a zero ($A \rightarrow Q$).

  • If $ax^{2} + x + a$ has equal zeroes, then $a = \frac{1}{2}$ or $a = \frac{-1}{2}$ ($B \rightarrow P$).

  • If $a+c=b$, then $-1$ is a zero ($C \rightarrow R$).

• Example 6: Constructing a Quadratic. If sum of zeroes = $-3$ and product = $2$, the polynomial is $x^{2} - (-3)x + 2 = x^{2} + 3x + 2$. General form: $k(x^{2} + 3x + 2)$.

• Example 7: Cubic Polynomial Verification. For $p(x) = 3x^{3} - 5x^{2} - 11x - 3$ with zeroes $3$, $-1$, $\frac{-1}{3}$.

  • $p(3) = 81 - 45 - 33 - 3 = 0$.

  • $\alpha + \beta + \gamma = 3 - 1 - \frac{1}{3} = \frac{5}{3} = \frac{-(-5)}{3}$.

  • $\alpha\beta\gamma = 3 \times (-1) \times (\frac{-1}{3}) = 1 = \frac{-(-3)}{3}$.

Questions & Discussion

• Exercise 2.1: Assessing the number of zeroes by looking at the number of points of intersection/touching on the $x$-axis. For instance, a line crossing only once has 1 zero; a curve crossing the x-axis four times has 4 zeroes.

• Real-world application (Bridge Hanging Wire): The shape of the hanging wire on a bridge is a parabola. This represents a quadratic polynomial of the form $y = ax^{2} + bx + c$. If the sum of zeroes for such a wire is $-3$ and the product is $5$, the expression for the hanging wire shape is $x^{2} + 3x + 5$.

• Composite and Prime Note: The transcript includes a handwritten note about composite numbers (having more than two factors, e.g., 4, 6, 8, 9, 10, 12, 14) and 1 being neither prime nor composite.

• Formula for nth degree: If $p(x)$ is of degree $n$, then the number of zeroes is less than or equal to $n$.