quick math factorization
Enter expression in this form
Example: 3x^2 + 9x
Factorization
Factorization is the process of breaking down an algebraic expression into a product of simpler expressions, or factors, which when multiplied together give the original expression. This is a crucial skill in algebra, as it simplifies expressions and solves equations more easily.
Types of Factorization
1. Common Factor: Taking out the greatest common factor (GCF) from terms.
2. Difference of Squares: Expressions in the form \(a^2 – b^2\), which factorizes to \((a – b)(a + b)\).
3. Quadratic Trinomials: Quadratics in the form \(ax^2 + bx + c\), which factorize to \((mx + n)(px + q)\).
4. Perfect Square Trinomials: Expressions like \(a^2 + 2ab + b^2\), which factorize to \((a + b)^2\).
Detailed Example
Let’s factorize the quadratic expression \(x^2 + 5x + 6\).
Steps:
1. Identify the coefficients:
– Here, \(a = 1\), \(b = 5\), and \(c = 6\).
2. Find two numbers that multiply to \(c\) (constant term) and add up to \(b\) (coefficient of \(x\)):
– We need two numbers that multiply to \(6\) and add to \(5\).
– These numbers are \(2\) and \(3\) because \(2 \times 3 = 6\) and \(2 + 3 = 5\).
3. Write the expression in factored form using the two numbers found:
– \(x^2 + 5x + 6\) can be written as \((x + 2)(x + 3)\).
4. Verify by expanding:
– Multiply the factors to ensure they give the original expression:
\[(x + 2)(x + 3) = x(x + 3) + 2(x + 3)\]
\[= x^2 + 3x + 2x + 6\]
\[= x^2 + 5x + 6\]
The original expression \(x^2 + 5x + 6\) is correctly factorized as \((x + 2)(x + 3)\).
