Multiplication of Functions Flashcards

Polynomial Expressions

• Polynomials are mathematical expressions involving variables raised to whole number powers.
• Each polynomial is composed of terms that include coefficients, variables, and exponents.
• Degree of a polynomial is determined by the highest power of the variable present.

Examples of Polynomial Forms

• -88x³ + 48x² + 88x - 48: A cubic polynomial with coefficients -88, 48, 88, and -48.
• -35k³ + 70k² + 15k - 30: Another cubic polynomial whose terms reflect similar structures.
• 32m³ - 16m² + 24m - 12: This polynomial has a cubic term, a quadratic term, a linear term, and a constant.
• 20x³ - 18x² - 50x + 45: Features a cubic term that dominates the polynomial, highlighting its non-linear nature.
• -40b³ + 55b² + 16b - 22: A cubic polynomial that illustrates a variety of coefficients.
• 45r³ + 50r² + 63r + 70: Exhibits a positive leading term, indicating growth behavior as r increases.

Function Multiplication

• Functions can be multiplied together to create a new polynomial expression representing their combined behavior.
• Example f(x) = 10x - 3 and g(x) = 6x + 8 leads to (fg)(x) = 60x² + 62x - 24.
• Example f(x) = 2x - 9 and g(x) = 4x + 5 results in (fg)(x) = 8x² - 26x - 45.

Additional Function Combinations

• (fg)(x) for polynomials like f(x) = 5x - 1 and g(x) = 5x + 3 yields 25x² + 10x - 3.
• Combining f(x) = 2x - 3 and g(x) = 4x + 5 gives (fg)(x) = 8x² - 2x - 15, showing interaction between linear terms.
• Each combination reflects how changing coefficients influences the final product.

Evaluating Function Products

• Determining (fg)(x) involves applying distribution across the terms in f(x) and g(x).
• Understanding polynomial multiplication is critical for grasping advanced algebra concepts.
• Each resultant polynomial showcases a unique combination of structural components from the original functions.