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Chapter 6: Problem 26

Divide and check. $$ \left(a^{2}-81\right) \div(a-9) $$

Short Answer

Expert verified
a + 9

Step by step solution

01

Identify the Expression

We need to divide \(a^2 - 81\) by \(a - 9\).
02

Simplify the Numerator

Notice that \(a^2 - 81\) is a difference of squares. Recall that \(x^2 - y^2 = (x+y)(x-y)\). Here, \(a^2 - 81 = (a)^2 - (9)^2\), so it simplifies to \(a^2 - 81 = (a + 9)(a - 9)\).
03

Divide the Simplified Expression

Now, divide \(a^2 - 81\) by \(a - 9\). In the simplified form, it becomes \(\frac{(a + 9)(a - 9)}{a - 9}\). Cancel out the common factor \(a - 9\) from the numerator and the denominator. The result is \(a + 9\).
04

Check the Solution

To ensure the solution is correct, multiply \(a + 9\) by \(a - 9\) and check if it expands back to the original expression \(a^2 - 81\). Indeed, \( (a + 9)(a - 9) = a^2 - 9a + 9a - 81 = a^2 - 81 \), which confirms our division is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The first thing to notice in this problem is that we are dealing with a special kind of polynomial known as a difference of squares. A difference of squares is a term like \(a^2 - b^2\) which can be factored into \((a + b)(a - b)\).

This is useful because it turns a single complicated expression into a product of simpler expressions. Knowing this formula will help you recognize and factor polynomials swiftly.

For example, in our problem, we have \(a^2 - 81\). Since 81 is a perfect square (\(9^2 = 81\)), we can rewrite \(81\) as \(9^2\). So, \(a^2 - 81\) becomes \(a^2 - 9^2\). This fits the difference of squares pattern.
Factoring Polynomials
Once you recognize that you have a difference of squares, the next step is to factor the polynomial.

Using our formula from earlier, we write \(a^2 - 81\) as \(a^2 - 9^2\), which factors to \((a + 9)(a - 9)\).

Factoring polynomials is breaking down a complicated polynomial into simpler 'factor' polynomials that, when multiplied together, will give back the original polynomial. This step is crucial because it simplifies the division process.

In this exercise, we simplified \(a^2 - 81\) to \((a + 9)(a - 9)\), which makes it easier to divide by \(a - 9\) in the next steps.
Simplification
Now that we have factored \(a^2 - 81\) into \((a + 9)(a - 9)\), dividing by \(a - 9\) becomes straightforward.
We express the division as \(\frac{(a + 9)(a - 9)}{a - 9}\).

In this fraction, the \(a - 9\) terms cancel each other out, leaving us with just \(a + 9\). This step is known as simplification.

Simplification is about reducing an expression to its simplest form. It involves canceling out common factors in the numerator and the denominator of a fraction.

To check if our simplification is correct, we can multiply \(a + 9\) by \(a - 9\) to see if we get back to \(a^2 - 81\). Indeed, \((a + 9)(a - 9) = a^2 - 81\), confirming that our division was done correctly.

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