- Greatest Common Factor: The Greatest Common Factor (GCF) is the largest number which can divide a set of numbers simultaneously.
- You know you have reached the GCF when there is no other number that can divide the set of numbers you are working with simultaneously.
- Polynomials: A polynomial refers to an expression with two or more terms.
Examples of polynomials and their GCF:
- 4x4 + 2x2 + x = x(4x3 + 2x + 1)
- y5 + 6y3 + y2 = y2(y3 + 6y + 1)
- z4 + 2z3 + z2 + 7z = z(z3 + 2z2 + z + 7)
- Rational Expression: Rational expression refers to the quotient of two polynomials.
- Explanation of rational expression: Rational expressions in polynomials are similar to rational numbers in integers.
Examples of rational expressions and their simplified forms:
- 4x + 8/ 4x – 8 = 4(x + 2)/ 4(x – 2) = x + 2/x – 2
- 6z2 + 3z/ 3z2 + 3z = 3z(2z + 1)/3z(z + 3) = 2z + 1/ z + 3
- Factoring the difference of two squares: The model used to factor the difference of two squares is as follows:
a2 – b2 = (a + b) (a – b)
- Factoring perfect square trinomials: The models used to factor perfect square polynomials is as follows:
a2 + 2ab + b2 = (a + b) 2 and a2 - 2ab + b2 = (a - b) 2
- All of the above models make sense to me.
- Factoring the difference of two squares seems much easier than factoring perfect square trinomials because in factoring perfect square trinomials, you must first determine whether the expression is a perfect square trinomial. In contrast, the difference of two squares is obvious.
- The difference of two squares:
25x2 – 16y2 = (5x – 4y) (5x – 4y)
- Perfect square trinomial;
x2 + 24x + 144 = (x + 12)2