Multiplying Polynomials
To multiply the polynomials (3x² + 5y²) and (5x² - 3y²), follow the distributive property (also known as the FOIL method for binomials):
- Step 1: Distribute Each Term
- Multiply 3x² by both terms in the second polynomial.
- Multiply 5y² by both terms in the second polynomial.
- Step 2: Perform the Multiplications
- 3x² * 5x² = 15x⁴
- 3x² * (-3y²) = -9x²y²
- 5y² * 5x² = 25x²y²
- 5y² * (-3y²) = -15y⁴
- Step 3: Combine Like Terms
- Combine -9x²y² and 25x²y²:
- -9x²y² + 25x²y² = 16x²y²
- Final Result
- The product is: 15x⁴ + 16x²y² - 15y⁴
Factoring the Quadratic Equation
To factor the expression x² + 6x - 16, we will follow these steps:
- Step 1: Identify Coefficients
- The equation is in the form ax² + bx + c, where:
- a = 1
- b = 6
- c = -16
- Step 2: Find Two Numbers
- We need to find two numbers that multiply to ac (1 * -16 = -16) and add to b (6).
- The numbers 8 and -2 work because:
- 8 * (-2) = -16
- 8 + (-2) = 6
- Step 3: Rewrite the Equation
- Rewrite x² + 6x - 16 as:
- x² + 8x - 2x - 16
- Step 4: Factor by Grouping
- Group the terms:
- (x² + 8x) + (-2x - 16)
- Factor out common factors:
- x(x + 8) - 2(x + 8)
- Step 5: Final Factored Form
- The factored expression is:
- (x + 8)(x - 2)