Introduction to Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. They are foundational in mathematics, appearing in fields like physics (motion equations), economics (growth models), and engineering (signal processing). Polynomials range from simple linear expressions to complex high-degree curves, and their study involves understanding their structure, operations, graphical behavior, and real-world applications. This guide provides a comprehensive overview with examples, operations like division and factoring, graphical insights, and practical uses to enhance your understanding.

Definition

A polynomial in one variable \( x \) is expressed as:

\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]

Where:

  • \( n \): Degree (highest exponent, a non-negative integer)
  • \( a_n, a_{n-1}, \ldots, a_0 \): Coefficients (real numbers, with \( a_n \neq 0 \))
  • \( a_n x^n \): Leading term
  • \( a_0 \): Constant term

Example: \( 3x^2 + 2x - 1 \) (degree 2, leading coefficient 3, constant term -1). Polynomials can have multiple variables, e.g., \( 2x^2 y + 3xy - 5 \), but we focus on single-variable polynomials here. The degree determines the polynomial’s end behavior and number of roots (up to \( n \) for real roots).

Types of Polynomials

  • Monomial: Single term, e.g., \( 5x^3 \) (degree 3).
  • Binomial: Two terms, e.g., \( x^2 - 4 \) (degree 2).
  • Trinomial: Three terms, e.g., \( x^2 + 3x + 2 \) (degree 2).
  • Quadratic: Degree 2, e.g., \( 2x^2 - 5x + 3 \).
  • Cubic: Degree 3, e.g., \( x^3 - 2x^2 + x - 1 \).
  • Quartic: Degree 4, e.g., \( x^4 - 3x^2 + 2 \).

Graphing Polynomials

The shape of a polynomial graph depends on its degree and leading coefficient. Let’s explore with examples.

Graph 1: Cubic Polynomial

Graph \( y = x^3 - 2x \) (degree 3, leading coefficient 1):

Degree 3 allows up to 3 x-intercepts; it rises on both ends (positive leading coefficient).

Graph 2: Quartic Polynomial

Graph \( y = x^4 - 4x^2 \) (degree 4, leading coefficient 1):

Degree 4 allows up to 4 x-intercepts; it rises on both ends.

Operations: Addition, Multiplication, Division, Factoring, Synthetic Division

Polynomials support various operations, crucial for simplification and solving.

Example 1: Addition

Add \( (2x^2 + 3x) + (x^2 - x) \):

\[ (2x^2 + 3x) + (x^2 - x) \] \[ = (2x^2 + x^2) + (3x - x) \] \[ = 3x^2 + 2x \]

Example 2: Multiplication

Multiply \( (x + 2)(x - 3) \):

\[ (x + 2)(x - 3) \] \[ = x \cdot x + x \cdot (-3) + 2 \cdot x + 2 \cdot (-3) \] \[ = x^2 - 3x + 2x - 6 \] \[ = x^2 - x - 6 \]

Example 3: Division (Long Division)

Divide \( x^3 - 6x^2 + 11x - 6 \) by \( x - 2 \):

\[ x - 2 \text{ into } x^3 - 6x^2 + 11x - 6 \] \[ x^3 \div x = x^2 \] \[ x^2 (x - 2) = x^3 - 2x^2 \] \[ (x^3 - 6x^2 + 11x - 6) - (x^3 - 2x^2) = -4x^2 + 11x - 6 \] \[ -4x^2 \div x = -4x \] \[ -4x (x - 2) = -4x^2 + 8x \] \[ (-4x^2 + 11x - 6) - (-4x^2 + 8x) = 3x - 6 \] \[ 3x \div x = 3 \] \[ 3 (x - 2) = 3x - 6 \] \[ (3x - 6) - (3x - 6) = 0 \]

Quotient: \( x^2 - 4x + 3 \), remainder: 0.

Example 4: Factoring

Factor \( x^2 + 5x + 6 \):

\[ x^2 + 5x + 6 \] \[ = (x + 2)(x + 3) \]

Verify: \( (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 \).

Example 5: Synthetic Division

Divide \( x^3 - 2x^2 + 3x - 4 \) by \( x - 1 \):

\[ \text{Coefficients: } 1, -2, 3, -4 \] \[ \text{Root: } 1 \] \[ 1 \mid 1 \ -2 \ 3 \ -4 \] \[ \quad \ 1 \ -1 \ 2 \ 0 \] \[ \quad 1 \ -3 \ 5 \ -4 \] \[ \text{Quotient: } x^2 - 3x + 5, \text{ Remainder: } -4 \]

Applications

Polynomials model diverse real-world scenarios. Here are detailed examples with calculations:

  • Motion - Projectile Height: Height \( s(t) = -16t^2 + 40t + 2 \) (feet, seconds). When does it hit the ground?
    \[ -16t^2 + 40t + 2 = 0 \] \[ \text{Multiply by -1: } 16t^2 - 40t - 2 = 0 \] \[ t = \frac{-(-40) \pm \sqrt{(-40)^2 - 4(16)(-2)}}{2(16)} \] \[ t = \frac{40 \pm \sqrt{1600 + 128}}{32} \] \[ t = \frac{40 \pm \sqrt{1728}}{32} \] \[ \sqrt{1728} \approx 41.57 \] \[ t = \frac{40 \pm 41.57}{32} \]

    Solutions:

    \[ t = \frac{40 + 41.57}{32} \approx \frac{81.57}{32} \approx 2.55 \] \[ t = \frac{40 - 41.57}{32} \approx \frac{-1.57}{32} \approx -0.05 \]

    Hits ground at \( t \approx 2.55 \) seconds.

  • Financial Growth: Revenue \( R(x) = -0.01x^2 + 10x \) (dollars, units sold). Maximize revenue.
    \[ x = -\frac{10}{2(-0.01)} \] \[ x = 500 \] \[ R = -0.01(500)^2 + 10(500) \] \[ = -2500 + 5000 \] \[ = 2500 \]

    Maximum revenue is $2500 at 500 units.

  • Engineering - Bridge Design: Deflection \( d(x) = 0.001x^4 - 0.1x^2 + 5 \) (meters). Find deflection at \( x = 10 \).
    \[ d(10) = 0.001(10)^4 - 0.1(10)^2 + 5 \] \[ = 0.001(10000) - 0.1(100) + 5 \] \[ = 10 - 10 + 5 \] \[ = 5 \]

    Deflection is 5 meters.

  • Agriculture - Crop Yield: Yield \( Y(x) = -0.02x^2 + 3x \) (tons, acres). Maximize yield for \( x \leq 50 \).
    \[ x = -\frac{3}{2(-0.02)} \] \[ x = 75 \] \[ \text{Since } 75 > 50, \text{ use } x = 50 \] \[ Y(50) = -0.02(50)^2 + 3(50) \] \[ = -50 + 150 \] \[ = 100 \]

    Maximum yield is 100 tons at 50 acres (constrained).

  • Polynomial Interpolation: Fit a quadratic through points (0, 1), (1, 3), (2, 7). Assume \( y = ax^2 + bx + c \).
    \[ 1 = c \quad (x = 0) \] \[ 3 = a + b + 1 \quad (x = 1) \] \[ 7 = 4a + 2b + 1 \quad (x = 2) \] \[ 2 = a + b \] \[ 6 = 4a + 2b \] \[ 6 - 2(2) = 4a + 2b - 2a - 2b \] \[ 2 = 2a \] \[ a = 1 \] \[ 2 = 1 + b \] \[ b = 1 \] \[ c = 1 \]

    Polynomial: \( y = x^2 + x + 1 \).

  • Physics - Pendulum Motion: Displacement \( d(t) = 0.1t^3 - 2t^2 + 5t \) (cm). Find displacement at \( t = 3 \).
    \[ d(3) = 0.1(3)^3 - 2(3)^2 + 5(3) \] \[ = 0.1(27) - 2(9) + 15 \] \[ = 2.7 - 18 + 15 \] \[ = -0.3 \]

    Displacement is -0.3 cm.