Polinomios: Monomios, Binomios y Trinomios

• Polynomials are algebraic expressions containing one or more terms, with each term including a variable raised to a non-negative integer power
• Terms are separated by addition or subtraction

Monomials

• Monomials consist of a single term
• Examples:
  • 5x
  • 3
  • 7a^2

Binomials

• Binomials consist of two terms
• Examples:
  • x + 2
  • 3y - 5
  • a^2 + b^2

Trinomials

• Trinomials consist of three terms
• Examples:
  • x^2 + 3x + 1
  • 2a - b + c
  • 4p^2 - q + 9

Adding Polynomials

• Combine like terms (terms with the same variable and exponent)
• Add the coefficients of like terms
• Example: (3x^2 + 2x - 1) + (x^2 - x + 4) = (3x^2 + x^2) + (2x - x) + (-1 + 4) = 4x^2 + x + 3

Subtracting Polynomials

• Distribute the negative sign to each term in the polynomial being subtracted
• Combine like terms
• Subtract the coefficients of like terms
• Example: (5x^2 - 3x + 2) - (2x^2 + x - 1) = 5x^2 - 3x + 2 - 2x^2 - x + 1 = (5x^2 - 2x^2) + (-3x - x) + (2 + 1) = 3x^2 - 4x + 3

Multiplying Polynomials

• Distribute each term in the first polynomial to each term in the second polynomial
• Combine like terms
• FOIL method (First, Outer, Inner, Last) can be used for multiplying two binomials
• Example: (x + 2)(x - 3) = x(x) + x(-3) + 2(x) + 2(-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6

Multiplying Polynomials - Distributive Property

• Multiply each term of one polynomial by each term of the other polynomial, then simplify
• Example: 2x(x^2 + 3x - 4) = 2x * x^2 + 2x * 3x - 2x * 4 = 2x^3 + 6x^2 - 8x

Multiplying Polynomials - Two Binomials

• Multiply each term in the first binomial by each term in the second
• Combine like terms
• Example: (x+3)(x+2) = xx + x2 + 3x + 32 = x^2 + 2x + 3x + 6 = x^2 + 5x + 6

Multiplying Polynomials - Polynomials with More Terms

• Distribute each term of the first polynomial to every term of the second polynomial
• Example: (x + 2)(x^2 - 3x + 5) = x(x^2) + x(-3x) + x(5) + 2(x^2) + 2(-3x) + 2(5) = x^3 - 3x^2 + 5x + 2x^2 - 6x + 10 = x^3 - x^2 - x + 10

Dividing Polynomials

• Polynomial long division: A method for dividing a polynomial by another polynomial of lower or equal degree
• Synthetic division: A shortcut method for dividing a polynomial by a linear binomial of the form (x - a)

Dividing Polynomials - Long Division

• Similar to long division with numbers
• Divide the term with the highest power in the dividend by the term with the highest power in the divisor
• Multiply the result by the divisor, subtract from the dividend, and bring down the next term
• Repeat until no more terms can be brought down
• Example: Divide (x^2 + 5x + 6) by (x + 2)
  • x + 3
  • x + 2 | x^2 + 5x + 6
  • -(x^2 + 2x)
  • 3x + 6
  • -(3x + 6)
  • 0
• Result: x + 3

Dividing Polynomials - Synthetic Division

• Used when dividing by a linear factor (x - a)
• Write down the coefficients of the dividend and the value of 'a'
• Bring down the first coefficient, multiply by 'a', add to the next coefficient, and repeat
• The last number is the remainder
• Example: Divide (x^2 + 5x + 6) by (x + 2)
  • -2 | 1 5 6
  • | -2 -6
  • | 1 3 0
• Result: x + 3

Este material explora los polinomios, incluyendo monomios, binomios y trinomios. Aprende a identificar y clasificar diferentes tipos de polinomios según su número de términos. También cubre cómo sumar y restar polinomios combinando términos semejantes.