Factoring Quadratic Expressions and Difference of Squares

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8 Questions

What is the general form of a quadratic expression?

ax^2 + bx + c

What is the formula for the difference of squares?

a^2 - b^2 = (a + b)(a - b)

What is the name of the method used to factor expressions involving the sum or difference of cubes?

Sum and difference formulas

What is the first step in polynomial long division?

Write the dividend and divisor

What is the purpose of factoring quadratic expressions?

To express the expression as a product of two binomials

What is the name of the method used to factor expressions by rearranging the terms to create common factors?

Factorising by grouping

What is the result of factoring the expression x^2 - 4?

(x + 2)(x - 2)

What is the final step in polynomial long division?

Write the final result in quotient and remainder form

Study Notes

Factoring Quadratic Expressions

  • 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)

Difference Of Squares

  • A difference of squares is a quadratic expression of the form: a^2 - b^2
  • Formula: a^2 - b^2 = (a + b)(a - b)
  • Example: x^2 - 4 = (x + 2)(x - 2)

Sum And Difference Formulas

  • 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

  • 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

  • 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

Quadratic Expressions

  • 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)

Factoring Methods

  • Factoring out the greatest common factor (GCF)
  • ac method: finding two numbers whose product is ac and whose sum is b

Difference of Squares

  • Defined as a quadratic expression of the form: a^2 - b^2
  • Formula: a^2 - b^2 = (a + b)(a - b)
  • Example: x^2 - 4 = (x + 2)(x - 2)

Sum and Difference Formulas

  • 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)
  • Used to factor expressions involving the sum or difference of cubes

Factorising by Grouping

  • Involves rearranging terms of a polynomial to create common factors
  • Example: 2x^2 + 5x + 3x + 7 = (x + 1)(2x + 7)

Polynomial Long Division

  • 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

Learn about quadratic expressions, factoring methods, and difference of squares. Understand the concepts of greatest common factor and ac method.

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