Mathematics Questions and Examples

1. Greatest Common Factor: The Greatest Common Factor (GCF) is the largest number which can divide a set of numbers simultaneously.
2. 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.
3. 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)

1. Rational Expression: Rational expression refers to the quotient of two polynomials.
2. 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

1. 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)

1. 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  

1. All of the above models make sense to me.
2. 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.

Examples:

• The difference of two squares:

25x² – 16y² = (5x – 4y) (5x – 4y)

• Perfect square trinomial;

x² + 24x + 144 = (x + 12)²