Simplify (4-x)/(x^2+3x-28)
The question given, "Simplify (4-x)/(x^2+3x-28)," is a problem involving the simplification of a rational expression. The numerator of the expression is "4 - x," and the denominator is a quadratic expression "x^2 + 3x - 28." The task is to simplify this expression by factoring the quadratic polynomial in the denominator and then reducing the fraction to its simplest form by canceling out any common factors that appear in both the numerator and the denominator.
$\frac{4 - x}{x^{2} + 3 x - 28}$
Answer
Solution:
Step 1: Factor the quadratic expression
Step 1.1: Apply the AC method
Identify two numbers that multiply to give $ac$ (the product of the coefficient of $x^2$ and the constant term) and add up to $b$ (the coefficient of $x$). For the quadratic $x^2 + 3x - 28$, we need numbers that multiply to $-28$ and add to $3$. The numbers are $-4$ and $7$.
Step 1.2: Write the factors
The factored form of the denominator is $(x - 4)(x + 7)$, so the expression becomes $\frac{4 - x}{(x - 4)(x + 7)}$.
Step 2: Simplify the fraction
Step 2.1: Identify and cancel common factors
Notice that $4 - x$ is the negative of $x - 4$.
Step 2.1.1: Express 4 as $-(-4)$
Rewrite the numerator as $-(-4) - x$ to make the common factor more apparent.
Step 2.1.2: Factor out $-1$ from the numerator
Factor $-1$ from the numerator to get $-(-4 + x)$.
Step 2.1.3: Simplify the factored numerator
The numerator simplifies to $-(x - 4)$.
Step 2.1.4: Rearrange the terms
The numerator is now the same as the denominator term $x - 4$, but with a negative sign.
Step 2.1.5: Cancel the common factor
The term $x - 4$ cancels out from the numerator and denominator, leaving $-\frac{1}{x + 7}$.
Step 2.1.6: Write the final expression
The simplified expression is $-\frac{1}{x + 7}$.
Step 2.2: Adjust the negative sign
Place the negative sign in front of the fraction to get the final answer: $-\frac{1}{x + 7}$.
Knowledge Notes:
The AC method is a technique used to factor quadratic expressions of the form $ax^2 + bx + c$. It involves finding two numbers that multiply to $ac$ and add to $b$. Once these numbers are found, the quadratic can be factored into two binomials.
When simplifying rational expressions, it is important to look for common factors in the numerator and denominator that can be canceled. Remember that $a - b$ is the same as $-(b - a)$, which is a useful property when identifying common factors.
The final step in simplifying is to ensure that the expression is in its simplest form, with no common factors remaining other than 1, and the negative sign properly placed.
