Introduction
Factoring is a fundamental concept in algebra that involves breaking down a polynomial into its simplest form by finding its factors. In this article, we will focus on factoring the quadratic polynomial X^2 + 12X + 45 and determine its factored form.
Understanding Quadratic Equations
A quadratic equation is a second-degree polynomial equation that consists of terms raised to the power of two (X^2). The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. To find the factored form of a quadratic equation, we need to factorize it into two binomial expressions.
Factoring X^2 + 12X + 45
To factor the quadratic polynomial X^2 + 12X + 45, we need to find two numbers that multiply to the constant term (45) and add up to the coefficient of the linear term (12). In this case, the numbers are 9 and 5, as 9 * 5 = 45 and 9 + 5 = 14. Therefore, we can rewrite the polynomial as:
- X^2 + 9X + 5X + 45
Next, we group the terms and factor by grouping:
- X(X + 9) + 5(X + 9)
Now, we can factor out the common factor (X + 9) from both terms:
- (X + 9)(X + 5)
Therefore, the factored form of the quadratic polynomial X^2 + 12X + 45 is (X + 9)(X + 5).
Additional Information
Factoring quadratic equations is a crucial skill in algebra, as it helps in solving various mathematical problems and simplifying expressions. Understanding the process of factoring polynomials allows us to manipulate equations efficiently and find solutions to complex problems.
- Factoring quadratics involves breaking down the polynomial into its simplest form by finding its factors.
- The factored form of a quadratic equation consists of two binomial expressions.
- To factor a quadratic polynomial, we need to identify two numbers that multiply to the constant term and add up to the coefficient of the linear term.
- Factoring by grouping involves separating the terms and factoring out the common factors.
Conclusion
In conclusion, the factored form of the quadratic polynomial X^2 + 12X + 45 is (X + 9)(X + 5). Factoring quadratic equations is a valuable skill that is essential for solving mathematical problems and simplifying expressions. By understanding the process of factoring polynomials, we can manipulate equations effectively and find solutions to various algebraic problems.
Remember to practice factoring quadratic equations to strengthen your algebraic skills and enhance your problem-solving abilities.
