Adding, Subtracting, and Multiplying Polynomial Expressions

Adding Polynomial Expressions

Example 1:

• Expression: (4x^2 + 5x + 7) + (3x^2 - 8x + 12)
• **Steps: **
  • Combine like terms:
    • 4x^2 + 3x^2 = 7x^2
    • 5x - 8x = -3x
    • 7 + 12 = 19
• Result: 7x^2 - 3x + 19

Example 2:

• Expression: (9x^2 - 7x + 13) - (5x^2 - 7x - 14)
• **Steps: **
  • Distribute negative sign to the second polynomial:
    • 9x^2 - 7x + 13 - 5x^2 + 7x + 14
  • Combine like terms:
    • 9x^2 - 5x^2 = 4x^2
    • -7x + 7x = 0
    • 13 + 14 = 27
• Result: 4x^2 + 27

Example 3:

• Expression: (3x^3 - 5x + 8 ) - (7x^2 + 6x - 9)
• **Steps: **
  • Distribute negative sign to the second polynomial:
    • 3x^3 - 5x + 8 - 7x^2 - 6x + 9
  • Combine like terms:
    • No like terms for 3x^3, so bring it down
    • Only one -7x^2 term
    • -5x - 6x = -11x
    • 8 + 9 = 17
• Result: 3x^3 - 7x^2 - 11x + 17

Multiplying Polynomial Expressions

Multiplying Two Binomials - FOIL Method

• Example: (3x + 5)(2x - 3)
• FOIL:
  • First: 3x * 2x = 6x^2
  • Outer: 3x * (-3) = -9x
  • Inner: 5 * 2x = 10x
  • Last: 5 * (-3) = -15
• Combine like terms:
  • -9x + 10x = x
• Result: 6x^2 + x - 15

Squaring a Binomial

• Example: (2x - 5)^2
• **Steps: **
  • Rewrite as (2x - 5)(2x - 5)
  • Apply FOIL:
    • First: 2x * 2x = 4x^2
    • Outer: 2x * (-5) = -10x
    • Inner: -5 * 2x = -10x
    • Last: -5 * (-5) = 25
  • Combine like terms:
    • -10x - 10x = -20x
• Result: 4x^2 - 20x + 25

Multiplying a Binomial by a Trinomial

• Example: (4x - 2)(x^2 + 3x - 5)
• Steps:
  • Distribute each term in the binomial:
    • 4x * x^2 = 4x^3
    • 4x * 3x = 12x^2
    • 4x * (-5) = -20x
    • -2 * x^2 = -2x^2
    • -2 * 3x = -6x
    • -2 * (-5) = 10
  • Combine like terms:
    • 4x^3 is unique
    • 12x^2 - 2x^2 = 10x^2
    • -20x - 6x = -26x
    • Constant term = 10
• Result: 4x^3 + 10x^2 - 26x + 10

Multiplying Two Trinomials

• Example: (3x^2 - 5x + 7)(2x^2 + 6x - 4)
• Steps:
  • Distribute each term:
    • 3x^2 * 2x^2 = 6x^4
    • 3x^2 * 6x = 18x^3
    • 3x^2 * (-4) = -12x^2
    • -5x * 2x^2 = -10x^3
    • -5x * 6x = -30x^2
    • -5x * (-4) = 20x
    • 7 * 2x^2 = 14x^2
    • 7 * 6x = 42x
    • 7 * (-4) = -28
  • Combine like terms:
    • 6x^4
    • 18x^3 - 10x^3 = 8x^3
    • -12x^2 - 30x^2 + 14x^2 = -28x^2
    • 20x + 42x = 62x
    • Constant term = -28
• Result: 6x^4 + 8x^3 - 28x^2 + 62x - 28

Dividing Polynomials

Using Factoring

• Example: (x^2 + 7x + 12) / (x + 3)
• Steps:
  • Factor numerator:
    • Find factors of 12 that add to 7: (3,4)
    • (x + 3)(x + 4)
  • Cancel out common terms:
    • Cancel (x + 3)
• Result: x + 4

Using Long Division

• Example 1: (2x^2 - x + 6) / (x - 2)

• **Steps: **

  • Divide 2x^2 by x: 2x
  • Multiply: 2x(x - 2) = 2x^2 - 4x
  • Subtract: (2x^2 - x + 6) - (2x^2 - 4x) = 3x
  • Bring down 6: 3x + 6
  • Divide 3x by x: 3
  • Multiply: 3(x - 2) = 3x - 6
  • Subtract: (3x + 6) - (3x - 6) = 0

• Result: 2x + 3

• Example 2: (2x^2 - 7x + 6) / (x - 2)

• Steps:

  • Divide 2x^2 by x: 2x
  • Multiply: 2x(x - 2) = 2x^2 - 4x
  • Subtract: (2x^2 - 7x + 6) - (2x^2 - 4x) = -3x + 6
  • Divide -3x by x: -3
  • Multiply: -3(x - 2) = -3x + 6
  • Subtract: (-3x + 6) - (-3x + 6) = 0

• Result: 2x - 3

Using Synthetic Division

• Example: (2x^2 - 7x + 6) / (x - 2)
• Steps:
  • Set denominator equal to 0: x - 2 = 0; x = 2
  • Write coefficients: 2 -7 6
  • Bring down first coefficient: 2
  • Multiply 2 by divisor: 2 * 2 = 4
  • Add to next coefficient: -7 + 4 = -3
  • Multiply result: -3 * 2 = -6
  • Add to next coefficient: 6 + (-6) = 0
• Result: 2x - 3