For all \(x>21, \frac{\left(x^{2}+8 x+7\right)(x-3)}{\left(x^{2}+4 x-21\right)(x+1)}=?\) F. 1 G. \(\frac{9}{7}\) H. \(\frac{x-3}{x+3}\) J. \(\frac{2(x-3)}{x+1}\) K. \(-\frac{4(x-3)}{x+1}\)

When approaching ACT Math practice problems that involve factoring quadratic expressions, it's essential to understand what 'factoring' actually means. Factoring is the process of decomposing a complex expression into a product of simpler factors. The quadratic expressions, in particular, take the form of ax^2 + bx + c.

One common method for factoring quadratics is to look for two numbers that multiply to give you the constant term (c) and add up to provide you with the linear coefficient (b). For the given exercise, we notice the quadratic expressions are already in the format where it is easy to identify these two numbers.

For example, for the quadratic expression x^2 + 8x + 7, we are looking for two numbers that multiply to 7 and add up to 8. In this case, 1 and 7 do the trick. Thus, this expression factors to (x+1)(x+7).

Visualizing Quadratic Factors

Sometimes picturing a rectangle with areas representing each term can help students conceptualize how these components fit together to build the expression. Moreover, understanding that factoring is the inverse of expanding can offer a pathway to check your work. If you multiply your factors and get back to the original quadratic expression, you've factored correctly!

Simplifying rational expressions is a common topic in ACT Math practice, which requires a strong grasp of both factoring and the properties of fractions. A rational expression is similar to a fraction, but instead of integers, you have polynomials in the numerator and denominator.

In the given exercise, we have a rational expression where both the numerator and the denominator contain factored quadratic expressions. Simplification involves dividing out any common factors that appear in both.

After factoring the quadratics, as demonstrated in the previous section, you’ll notice that both (x+1) and (x+7) appear in the numerator and the denominator. These can be cancelled out, simplifying our complex expression to the constant 1. This simplification embodies the key concept that anything divided by itself is 1, barring the scenario where that term is zero (as zero division is undefined).

Why Simplification Matters

The process of simplification not only makes the expressions more manageable but also prepares students for further algebraic operations, offering a clearer view of the relationships between algebraic quantities.

ACT Math practice problems will frequently test your ability to work with algebraic expressions. An algebraic expression is a mathematical statement that includes numbers, variables (like x or y), and operation signs. Algebraic expressions can range from simple, like x + 5, to more complex, like the one given in our exercise.

These expressions represent quantities that can vary, which is why they contain variables. The power of algebra comes from the fact that these variables can be manipulated through various operations to solve for unknowns or simplify expressions. A key aspect of working with algebraic expressions is following the fundamental rules of algebra, such as the distributive property, commutative property, and associative property.

Mastering algebraic expressions means being comfortable with variables and operations. It also means recognizing patterns, such as the difference or sum of cubes, or the square of a binomial. Understanding these concepts fundamentally will help students across a wide range of problems, including those not immediately obvious as algebraic in nature.