(x-2).jpeg "(x-6)(x-2)")
Understanding the Expression (x-6)(x-2)
The expression (x-6)(x-2) represents the product of two binomials. Let's break down what this means and how to simplify it:
Binomials and Multiplication
Binomials are algebraic expressions consisting of two terms. In this case, we have two binomials:
- (x-6)
- (x-2)
To multiply these binomials, we use the distributive property. This means we multiply each term in the first binomial by each term in the second binomial.
Expanding the Expression
Here's how we expand the expression:
- Multiply the first terms: x * x = x²
- Multiply the outer terms: x * -2 = -2x
- Multiply the inner terms: -6 * x = -6x
- Multiply the last terms: -6 * -2 = 12
Now, we combine all the terms:
x² - 2x - 6x + 12
Finally, we combine like terms:
x² - 8x + 12
Simplifying and Factoring
The simplified form of the expression (x-6)(x-2) is x² - 8x + 12. This is a quadratic expression, which means it's a polynomial with the highest power of x being 2.
It's important to remember that the original factored form, (x-6)(x-2), provides valuable information about the expression. It tells us that the expression can be factored into two linear binomials. This can be helpful when solving equations or analyzing the behavior of the function represented by the expression.
Applications
Understanding how to expand and simplify expressions like (x-6)(x-2) is crucial in various mathematical contexts, including:
- Solving equations: Factoring can help to find the roots of quadratic equations.
- Graphing functions: Understanding the factored form helps to identify the x-intercepts and the shape of the graph of a quadratic function.
- Calculus: The factored form can be used to find derivatives and integrals of functions.
In conclusion, the expression (x-6)(x-2) represents a product of two binomials. By expanding and simplifying it, we obtain a quadratic expression, x² - 8x + 12. This process and the resulting expressions are fundamental concepts in algebra and other branches of mathematics.