Question
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Answer
The result of the division is \( x - 7 \).
Solution
To perform the division \(\left(x^{2} - 4x - 21\right) \div (x + 3)\), you can use polynomial long division or factoring. Here's a straightforward method using factoring:
1. **Factor the numerator:**
The quadratic expression \(x^{2} - 4x - 21\) can be factored by finding two numbers that multiply to \(-21\) and add to \(-4\). These numbers are \(-7\) and \(3\).
\[
x^{2} - 4x - 21 = (x - 7)(x + 3)
\]
2. **Set up the division:**
\[
\frac{(x - 7)(x + 3)}{x + 3}
\]
3. **Simplify:**
You can cancel the \((x + 3)\) terms in the numerator and the denominator (provided \(x \neq -3\) to avoid division by zero):
\[
\frac{(x - 7)\cancel{(x + 3)}}{\cancel{x + 3}} = x - 7
\]
So, the result of the division is:
\[
x - 7
\]
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Simplify this solution Extra Insights
To solve \( \left(x^{2}-4 x-21\right) \div(x+3) \), let's first factor the quadratic expression. The expression \( x^2 - 4x - 21 \) factors into \( (x - 7)(x + 3) \). Now, when we divide by \( (x + 3) \), we can simplify the expression since \( x + 3 \) cancels out, giving us \( x - 7 \) for \( x \neq -3 \). This means that the solution to the division is \( x - 7 \), except at the point where \( x + 3 = 0 \) (or \( x = -3 \)), where the original expression is undefined. So you get a nice linear function with a hole! Want to try plugging in some values to see how it behaves?
