Algebra Class: Distributive Property and Methods

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Questions and Answers

What is the definition of Distributive?

  • x(y+z) = xy + xz (correct)
  • xy + xz = x(y + z)
  • a(b + c) = ab + ac
  • None of the above

What is x(y+z) simplified?

xy + xz

What operation does the FOIL method perform?

  • Factoring
  • Distributing
  • Subtracting
  • Multiplying two binomials (correct)

What is the result of (a+b)(x+y) using FOIL?

<p>ax + ay + bx + by</p> Signup and view all the answers

Simplify (x+a)(x+b).

<p>x^2 + (a+b)x + ab</p> Signup and view all the answers

What is the general form of a Quadratic Type 2 expression?

<p>(ax+b)(cx+d) = acx^2 + (ad+bc)x + bd</p> Signup and view all the answers

What is the simplified result of (x+y)^2?

<p>x^2 + 2xy + y^2</p> Signup and view all the answers

What is x^2 + (a+b)x + ab simplified?

<p>(x+a)(x+b)</p> Signup and view all the answers

Simplify (ax+b)(cx+d).

<p>acx^2 + (ad+bc)x + bd</p> Signup and view all the answers

What does Trinomial Perfect Square (x+y)^2 equal?

<p>x^2 + 2xy + y^2</p> Signup and view all the answers

What is (x+y)(x-y) simplified?

<p>x^2 - y^2</p> Signup and view all the answers

What is the result of (x+y+z)^2?

<p>x^2 + y^2 + z^2 + 2xy + 2yz + 2xz</p> Signup and view all the answers

What is the Cube of a Binomial (x+y)^3?

<p>x^3 + 3x^2y + 3xy^2 + y^3</p> Signup and view all the answers

What is the simplified result of (x-y)^3?

<p>x^3 - 3x^2y + 3xy^2 - y^3</p> Signup and view all the answers

What does the Sum of Two Cubes formula represent?

<p>x^3 + y^3 = (x+y)(x^2 - xy + y^2)</p> Signup and view all the answers

Simplify x^3 + y^3.

<p>(x+y)(x^2 - xy + y^2)</p> Signup and view all the answers

What is the Difference of Two Cubes formula?

<p>x^3 - y^3 = (x-y)(x^2 + xy + y^2)</p> Signup and view all the answers

What is (x-y)(x^2 + xy + y^2) simplified?

<p>x^3 - y^3</p> Signup and view all the answers

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Study Notes

Distributive Property

  • Represents the method of distributing multiplication over addition: (x(y+z) = xy + xz)
  • Essential for simplifying algebraic expressions.

Simplification Techniques

  • The expression (xy + xz) simplifies to (x(y + z)), showcasing the distributive property.
  • Understanding these simplifications aids in problem-solving and algebraic manipulation.

FOIL Method

  • FOIL stands for First, Outer, Inner, Last, used to expand the product of two binomials: ((a+b)(x+y) = ax + ay + bx + by)
  • An effective strategy for multiplying binomials.

Quadratic Expansions

  • Quadratic Type 1: ((x+a)(x+b) = x^2 + (a+b)x + ab)
  • Quadratic Type 2: ((ax+b)(cx+d) = acx^2 + (ad+bc)x + bd)
  • Quadratics represent polynomial expressions of the second degree.

Perfect Squares

  • Trinomial Perfect Square: The formula ((x+y)^2 = x^2 + 2xy + y^2) indicates how to expand the square of a binomial.
  • Another form: ((x-y)^2 = x^2 - 2xy + y^2) accounts for subtraction.

Difference of Squares

  • The formula ((x+y)(x-y) = x^2 - y^2) demonstrates how the product of a sum and difference results in a difference of squares.
  • This is a useful factorization technique.

Cubes of Binomials

  • Cube Formula for Sums: ((x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3)
  • Cube Formula for Differences: ((x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3)

Sum and Difference of Cubes

  • Sum of Two Cubes: (x^3+y^3 = (x+y)(x^2 - xy + y^2)) is crucial for factoring cubic expressions.
  • Difference of Two Cubes: (x^3-y^3 = (x-y)(x^2 + xy + y^2)) provides another factorization method.

Notations and Simplifications

  • Proficiency in recognizing and applying these formulas fosters stronger algebra skills.
  • Simplifying outcomes like (x^3+y^3) to ((x+y)(x^2-xy+y^2)) reinforces understanding of polynomial identities.

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