Real Numbers and Polynomials

TL;DR

You'll learn about different types of real numbers and how they fit together, then we'll dive into polynomials, understanding their structure and how to work with them, especially finding their "zeros."

1. The Mental Model

Think of real numbers as all the numbers you can place on a continuous line, without any breaks. Polynomials are like versatile "recipes" for creating numbers from variables, built with only addition, subtraction, and multiplication.

2. The Core Material

Understanding Real Numbers

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Real numbers are a big family of numbers. You already know some members!

• Natural Numbers (N): These are your counting numbers: 1, 2, 3, ...
• Whole Numbers (W): Just natural numbers plus zero: 0, 1, 2, 3, ...
• Integers (Z): Whole numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational Numbers (Q): Any number that can be written as a fraction p/q, where p and q are integers, and q is not zero. Examples: 1/2, -3/4, 5 (which is 5/1), 0.333... (which is 1/3).
• Irrational Numbers: Numbers that cannot be written as a simple fraction. Their decimal representation goes on forever without repeating. Examples: $\pi$ (pi), $\sqrt{2}$, $e$.

All these numbers together form the set of Real Numbers (R).

graph TD
    R["Real Numbers (All numbers on a number line)"]
    Q["Rational Numbers (p/q, decimals that terminate or repeat)"]
    I["Irrational Numbers (Decimals that never terminate or repeat)"]

    Z["Integers (..., -2, -1, 0, 1, 2, ...)"]
    W["Whole Numbers (0, 1, 2, ...)"]
    N["Natural Numbers (1, 2, 3, ...)"]

    R --> Q
    R --> I

    Q --> Z
    Z --> W
    W --> N

What are Polynomials?

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A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

For example:
* $P(x) = 3x^2 - 5x + 7$ is a polynomial in one variable, $x$.
* $Q(y) = y^3 + 2y - 1$
* $R(z) = 4z^5$
* Constants like $K(x) = 5$ are also polynomials (specifically, constant polynomials).

Key terms you should know:
* Terms: Parts of the polynomial separated by addition or subtraction. In $3x^2 - 5x + 7$, the terms are $3x^2$, $-5x$, and $7$.
* Coefficients: The numerical part of a term. In $3x^2$, the coefficient is $3$.
* Variable: The letter representing an unknown value, like $x$ or $y$.
* Exponent (or Degree of a term): The power to which the variable is raised. In $3x^2$, the exponent is $2$.
* Degree of a Polynomial: The highest exponent of the variable in the polynomial. For $3x^2 - 5x + 7$, the degree is $2$.
* Zeroes of a Polynomial: These are the values of the variable that make the polynomial equal to zero. If $P(x) = 0$, then $x$ is a zero. Also called roots. For example, if $P(x) = x - 2$, then $x=2$ is a zero because $P(2) = 2 - 2 = 0$.

Finding Zeros of Polynomials

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For simple polynomials:

• Linear Polynomials (Degree 1): $P(x) = ax + b$.
  To find the zero, set $P(x) = 0$: $ax + b = 0 \implies ax = -b \implies x = -b/a$.
  Example: $P(x) = 2x - 6$. Zero is $x = -(-6)/2 = 6/2 = 3$.

• Quadratic Polynomials (Degree 2): $P(x) = ax^2 + bx + c$.
  You can use factoring, completing the square, or the quadratic formula:
  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  The expression $b^2 - 4ac$ is called the discriminant ($\Delta$).

  • If $\Delta > 0$, there are two distinct real zeros.
  • If $\Delta = 0$, there is exactly one real zero (a repeated root).
  • If $\Delta < 0$, there are no real zeros (but two complex zeros, which you'll learn about later).

The Relationship Between Zeros and Coefficients (for quadratic polynomials)

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If $\alpha$ and $\beta$ are the zeros of $ax^2 + bx + c$:
* Sum of zeros: $\alpha + \beta = -b/a$
* Product of zeros: $\alpha \beta = c/a$

Division Algorithm for Polynomials

When you divide a polynomial $P(x)$ by another polynomial $G(x)$ (where $G(x)$ is not zero and its degree is less than or equal to $P(x)$'s degree), you get a quotient $Q(x)$ and a remainder $R(x)$.

$P(x) = G(x) \times Q(x) + R(x)$

Where the degree of $R(x)$ is less than the degree of $G(x)$, or $R(x) = 0$.
If $R(x) = 0$, then $G(x)$ is a factor of $P(x)$. This is crucial for finding zeros, especially when you have one zero and need to find others.

3. Worked Example

Let's find the zeros of the polynomial $P(x) = 2x^2 - 8x + 6$ and verify the relationship between its zeros and coefficients.

1. Identify coefficients: Here, $a=2$, $b=-8$, $c=6$.

2. Find the zeros using factoring:
   We need two numbers that multiply to $ac = 2 \times 6 = 12$ and add to $b = -8$. These numbers are $-2$ and $-6$.
   Rewrite the middle term: $2x^2 - 2x - 6x + 6 = 0$
   Factor by grouping: $2x(x - 1) - 6(x - 1) = 0$
   $(2x - 6)(x - 1) = 0$
   Set each factor to zero:
   $2x - 6 = 0 \implies 2x = 6 \implies x = 3$ (Let's call this $\alpha$)
   $x - 1 = 0 \implies x = 1$ (Let's call this $\beta$)
   The zeros are $x=3$ and $x=1$.

3. Verify the relationship between zeros and coefficients:

   • Sum of zeros: $\alpha + \beta = 3 + 1 = 4$.
     Using the formula: $-b/a = -(-8)/2 = 8/2 = 4$. (Matches!)
   • Product of zeros: $\alpha \beta = 3 \times 1 = 3$.
     Using the formula: $c/a = 6/2 = 3$. (Matches!)

This confirms our zeros are correct and the relationships hold true.

4. Key Takeaways

• Real numbers encompass natural, whole, integers, rational, and irrational numbers.
• Polynomials are expressions with variables and non-negative integer exponents, using only addition, subtraction, and multiplication.
• The degree of a polynomial is its highest exponent, and zeros are the values that make the polynomial equal to zero.
• For quadratic polynomials ($ax^2 + bx + c = 0$), zeros can be found by factoring or using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• The sum of zeros for $ax^2 + bx + c$ is $-b/a$, and their product is $c/a$.
• The Division Algorithm helps break down polynomials: $P(x) = G(x) \times Q(x) + R(x)$.

Common Mistakes to Avoid:
* Confusing rational numbers (like 1/3) with irrational numbers (like $\sqrt{3}$). Rational numbers have repeating or terminating decimals; irrational do not.
* Forgetting that the exponent in a polynomial must be a non-negative integer (no $\sqrt{x}$ or $x^{-1}$, for instance).
* Incorrectly applying the signs in the quadratic formula, especially with negative $b$ values.
* Assuming "zero" simply means the constant term; it's the value of the variable that makes the whole polynomial equal to zero.
* Ignoring the degree of the remainder in polynomial division; it must be less than the divisor's degree.

5. Now Try It

Consider the polynomial $P(x) = x^3 - 2x^2 - 5x + 6$. You are given that $x=1$ is one of its zeros. Use the Division Algorithm to divide $P(x)$ by $(x-1)$ to find the resulting quadratic polynomial. Then, find the other two zeros of $P(x)$ from that quadratic. Success looks like correctly performing the polynomial division and identifying all three zeros.