Factoring Quadratic Expressions and Difference of Squares

Study Notes

A quadratic expression is a polynomial of degree two, in the form: ax^2 + bx + c
Factoring quadratic expressions involves expressing them as a product of two binomials: (x + p)(x + q)
Methods for factoring quadratic expressions:
- Factoring out the greatest common factor (GCF)
- Looking for two numbers whose product is ac and whose sum is b (ac method)

Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
These formulas can be used to factor expressions involving the sum or difference of cubes

Factorising by grouping involves rearranging the terms of a polynomial to create common factors
Example: 2x^2 + 5x + 3x + 7 = (2x^2 + 5x) + (3x + 7) = x(2x + 5) + 1(3x + 7) = (x + 1)(2x + 7)

Polynomial long division is a method for dividing one polynomial by another
Similar to long division of numbers, but with polynomials instead
Steps:
1. Write the dividend (the polynomial being divided) and the divisor (the polynomial by which we are dividing)
2. Divide the leading term of the dividend by the leading term of the divisor
3. Multiply the result from step 2 by the divisor and subtract the product from the dividend
4. Repeat steps 2 and 3 until the degree of the remainder is less than the degree of the divisor
Used to factor polynomials, especially those that are not easily factored by other methods

Defined as polynomials of degree two, in the form: ax^2 + bx + c
Factoring involves expressing them as a product of two binomials: (x + p)(x + q)

Method for dividing one polynomial by another
Similar to long division of numbers, but with polynomials
Steps:
- Write dividend and divisor
- Divide leading term of dividend by leading term of divisor
- Multiply result by divisor and subtract product from dividend
- Repeat steps until degree of remainder is less than degree of divisor
Used to factor polynomials, especially those not easily factored by other methods