🔟Elementary Algebra Unit 6 – Polynomials

What Are Polynomials?

• Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication
• Formed by combining one or more terms, each term includes a coefficient and one or more variables raised to non-negative integer exponents
• General form of a polynomial: $a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0$, where $a_n, a_{n-1}, ..., a_0$ are coefficients and $x$ is the variable
  • Coefficients can be any real number, including zero
  • Exponents must be non-negative integers
• Degree of a polynomial determined by the highest power of the variable in any term
• Examples of polynomials: $3x^2 + 2x - 5$, $y^3 - 4y + 1$, $2a^4 - 3a^2b + ab^2 + 7$
• Non-examples of polynomials: $\frac{1}{x} + 3$ (negative exponent), $\sqrt{x} + 2$ (non-integer exponent), $|x| - 1$ (absolute value function)

Parts of a Polynomial

• Terms are the individual parts of a polynomial separated by addition or subtraction
  • Each term consists of a coefficient and a variable raised to a non-negative integer power
• Coefficients are the numeric factors multiplied by the variables in each term
  • Can be any real number, including positive, negative, or zero
• Variables are the letters or symbols representing unknown quantities in a polynomial
  • Most commonly represented by $x$, but can be any letter or symbol
• Exponents indicate the power to which the variable is raised in each term
  • Must be non-negative integers
• Leading term is the term with the highest degree in a polynomial
  • Coefficient of the leading term is called the leading coefficient
• Constant term is the term without any variables (degree 0)
• Example: In the polynomial $4x^3 - 2x^2 + 3x - 7$, there are 4 terms, the leading term is $4x^3$, the leading coefficient is 4, and the constant term is -7

Types of Polynomials

• Monomials are polynomials with only one term
  • Examples: $3x$, $-5y^2$, $2ab$
• Binomials are polynomials with exactly two terms
  • Examples: $x + 3$, $2y - 1$, $a^2 + b^2$
• Trinomials are polynomials with exactly three terms
  • Examples: $x^2 + 2x + 1$, $3y^2 - 5y + 2$, $a^3 + 2a^2 - 3a$
• Quadratic polynomials are polynomials of degree 2
  • General form: $ax^2 + bx + c$, where $a \neq 0$
• Cubic polynomials are polynomials of degree 3
  • General form: $ax^3 + bx^2 + cx + d$, where $a \neq 0$
• Quartic polynomials are polynomials of degree 4
  • General form: $ax^4 + bx^3 + cx^2 + dx + e$, where $a \neq 0$
• Polynomials can be classified by their degree (linear, quadratic, cubic, etc.) or by the number of terms (monomial, binomial, trinomial, etc.)

Operations with Polynomials

• Addition and subtraction of polynomials involve combining like terms
  • Like terms have the same variable raised to the same power
  • To add or subtract, keep the variable and exponent the same, and add or subtract the coefficients
  • Example: $(3x^2 + 2x - 1) + (2x^2 - 3x + 4) = 5x^2 - x + 3$
• Multiplication of polynomials involves distributing each term of one polynomial by every term of the other polynomial
  • Multiply the coefficients and add the exponents of like variables
  • Combine like terms after multiplying
  • Example: $(2x + 3)(x - 1) = 2x^2 - 2x + 3x - 3 = 2x^2 + x - 3$
• Division of polynomials can be performed using long division or synthetic division
  • Long division involves dividing each term of the dividend by the divisor and subtracting the result from the dividend
  • Synthetic division is a shortcut method for dividing a polynomial by a linear factor
• Polynomial long division example: $(x^3 + 2x^2 - 5x - 6) \div (x + 3) = x^2 - x - 2$
• Synthetic division example: Dividing $2x^3 - 3x^2 + 5x - 4$ by $x - 2$ results in a quotient of $2x^2 - x + 7$ and a remainder of 10

Factoring Polynomials

• Factoring is the process of expressing a polynomial as a product of its factors
• Common factoring techniques include:
  • Greatest common factor (GCF): Factor out the GCF of all terms
  • Difference of squares: $a^2 - b^2 = (a + b)(a - b)$
  • Perfect square trinomials: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$
  • Grouping: Group terms and factor out common binomial factors
  • Trinomial factoring: Factor trinomials of the form $ax^2 + bx + c$ by finding factors of $ac$ that add up to $b$
• Factoring is useful for simplifying expressions, solving equations, and finding roots of polynomials
• Example: Factoring $3x^2 + 8x - 3$ by trinomial factoring: $(3x - 1)(x + 3)$
• Example: Factoring $x^4 - 16$ using difference of squares: $(x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2)$

Solving Polynomial Equations

• To solve a polynomial equation, set the polynomial equal to zero and find the values of the variable that make the equation true
• For linear equations (degree 1), isolate the variable on one side of the equation and solve
  • Example: $2x - 5 = 0$, add 5 to both sides, then divide by 2 to get $x = \frac{5}{2}$
• For quadratic equations (degree 2), use factoring, the quadratic formula, or completing the square
  • Factoring: Set the quadratic equal to zero, factor, and set each factor equal to zero to find the solutions
  • Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for a quadratic equation in the form $ax^2 + bx + c = 0$
  • Completing the square: Rewrite the quadratic in the form $(x + p)^2 = q$ and solve for $x$
• For higher-degree polynomials, use factoring, graphing, or numerical methods to approximate solutions
• Example: Solving $x^2 - 5x + 6 = 0$ by factoring: $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$
• Example: Solving $2x^2 + x - 3 = 0$ using the quadratic formula: $x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-3)}}{2(2)} = \frac{-1 \pm \sqrt{25}}{4} = \frac{-1 \pm 5}{4}$, so $x = 1$ or $x = -\frac{3}{2}$

Graphing Polynomials

• Polynomial functions can be graphed on a coordinate plane
• Key features of polynomial graphs include:
  • Degree: Determines the maximum number of turning points (degree - 1) and the end behavior
  • Leading coefficient: Determines the direction of the end behavior (positive = up, negative = down)
  • Zeros (roots): Points where the graph crosses the x-axis ($y = 0$)
  • Y-intercept: Point where the graph crosses the y-axis ($x = 0$)
• To graph a polynomial, plot points or use transformations of parent functions
• Quadratic functions (degree 2) have a parabolic shape and can be graphed using the vertex form $y = a(x - h)^2 + k$
  • Vertex is the turning point of the parabola, located at $(h, k)$
  • Axis of symmetry is the vertical line that passes through the vertex
• Higher-degree polynomials may have more complex shapes with multiple turning points
• Example: Graphing $y = x^3 - 4x$, a cubic function with zeros at $x = -2$, $x = 0$, and $x = 2$, and a y-intercept at $(0, 0)$
• Example: Graphing $y = -2(x + 1)^2 + 3$, a quadratic function with vertex at $(-1, 3)$ and axis of symmetry at $x = -1$

Real-World Applications

• Polynomials are used to model various real-world situations in fields such as physics, engineering, economics, and more
• Quadratic functions can model projectile motion, with the height of the object as a function of time
  • Example: The height $h$ (in meters) of a ball thrown upward with an initial velocity of 20 m/s from a height of 1.5 meters can be modeled by $h(t) = -4.9t^2 + 20t + 1.5$
• Cubic functions can model the volume of a box with a square base and a fixed surface area
  • Example: The volume $V$ (in cubic centimeters) of a box with a square base and a surface area of 150 square centimeters can be modeled by $V(x) = x(75 - 2x^2)$, where $x$ is the side length of the base
• Polynomial regression can be used to fit a polynomial function to a set of data points
  • Example: Given data points $(1, 3)$, $(2, 6)$, $(3, 11)$, and $(4, 18)$, a quadratic regression model can be found: $y = x^2 + x + 1$
• Polynomials are used in computer graphics to create smooth curves and surfaces through techniques like Bézier curves and splines
• In physics, polynomials are used to describe the motion of objects, such as the position, velocity, and acceleration of a particle over time