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Polynomials and Factoring

1️⃣ What is a Polynomial?

A polynomial is an algebraic expression consisting of variables, exponents, and coefficients combined using addition, subtraction, and multiplication.

Key Features of a Polynomial:

  • Each term consists of a coefficient, a variable, and an exponent.
  • Polynomials cannot have variables in denominators, negative exponents, or square roots of variables.
  • The degree of a polynomial is the highest exponent of \( x \).

Examples of Polynomials:

  • \( 3x^2 - 4x + 5 \) (Quadratic polynomial)
  • \( x^3 + 2x^2 - x + 6 \) (Cubic polynomial)
  • \( 5x - 7 \) (Linear polynomial)

Not Polynomials:

  • \( \frac{3}{x} + 2 \) (variable in denominator)
  • \( x^{-2} + 4x \) (negative exponent)
  • \( \sqrt{x} + 2x \) (square root of a variable)

2️⃣ Adding and Subtracting Polynomials

To add or subtract polynomials, combine like terms (terms with the same variable and exponent).

Example 1: Adding

\[ (2x^2 + 3x + 4) + (x^2 - 5x + 6) \]
Combine like terms: \( (2x^2 + x^2) + (3x - 5x) + (4 + 6) \)
Final Answer: \( 3x^2 - 2x + 10 \)

Example 2: Subtracting

\[ (4x^2 - 2x + 7) - (3x^2 + 5x - 4) \]
Distribute the negative: \( 4x^2 - 2x + 7 - 3x^2 - 5x + 4 \)
Combine like terms: \( (4x^2 - 3x^2) + (-2x - 5x) + (7 + 4) \)
Final Answer: \( x^2 - 7x + 11 \)

3️⃣ Multiplying Polynomials

Multiplying a Monomial by a Polynomial

Example: Multiply \( 3x(2x^2 - 4x + 5) \)

Distribute \( 3x \) to each term: \[ 3x \cdot 2x^2 + 3x \cdot (-4x) + 3x \cdot 5 = 6x^3 - 12x^2 + 15x \]
Final Answer: \( 6x^3 - 12x^2 + 15x \)

Multiplying Binomials Using the FOIL Method

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

FOIL: First, Outer, Inner, Last: \[ x \cdot x + x \cdot (-2) + 3 \cdot x + 3 \cdot (-2) \]
Simplify: \( x^2 - 2x + 3x - 6 = x^2 + x - 6 \)

4️⃣ Factoring Polynomials

Factoring Quadratic Trinomials

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

Find two numbers that multiply to 6 and add to 5: (-2 and -3) when solving an equation, you’d set: \[ (x + 2)(x + 3) = 0 \]
Solution (if solving): \( x = -2 \) or \( x = -3 \)

Factoring When \( a \neq 1 \)

Example: Factor \( 2x^2 + 5x + 3 \)

Find two numbers that multiply to \( 2 \times 3 = 6 \) and add to 5: (2 and 3).
Rewrite: \( 2x^2 + 2x + 3x + 3 \)
Factor by grouping: \( 2x(x+1) + 3(x+1) = (2x+3)(x+1) \)

Special Factoring Cases

Difference of Squares: \( a^2 - b^2 = (a - b)(a + b) \)
Example: Factor \( x^2 - 16 \) → \( (x - 4)(x + 4) \)

Perfect Square Trinomials: \( a^2 + 2ab + b^2 = (a + b)^2 \)
Example: Factor \( x^2 + 6x + 9 \) → \( (x + 3)^2 \)

📌 Practice Questions

Basic Practice

  1. Identify the degree of \( x^3 - 2x^2 + 4x - 7 \).
  2. Classify \( 4x^2 + 5x - 3 \) by terms and degree.
  3. Add \( (x^2 + 3x - 4) + (2x^2 - 5x + 7) \).
  4. Subtract \( (4x^3 - 2x + 6) - (x^3 + 5x - 8) \).
  5. Multiply \( (3x + 4)(x - 2) \) using the distributive property.
  6. Factor \( x^2 + 7x + 10 \).
  7. Factor \( 4x^2 - 25 \) using the difference of squares.

Challenging Questions

  1. The sum of two polynomials is \( 5x^2 + 7x - 9 \). If one polynomial is \( 2x^2 - 4x + 5 \), find the other.
  2. A rectangle has length \( (3x + 2) \) and width \( (x - 1) \). Find its perimeter.
  3. A square has side length \( (x + 2) \). Find its area.
  4. Factor and solve \( 5x^2 - 13x + 6 = 0 \).

7️⃣ Summary

  • ✅ Polynomials are expressions with variables, exponents, and coefficients.
  • ✅ Adding, subtracting, and multiplying polynomials involves combining like terms.
  • ✅ Factoring rewrites polynomials as products of simpler expressions.
  • ✅ Recognizing special patterns (difference of squares, perfect square trinomials) speeds up factoring.

🎉 Great job!

Next topic: Introduction to Quadratic Equations! 🚀

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