Problem 101 - Elementary Algebra

Question

For the following problems, divide the polynomials. $$ a^{2}+9 a+18 \text { by } a+3 $$

Step-by-Step Solution

Answer

Question: Divide the polynomial $$a^2 + 9a + 18$$ by the linear term $$a+3$$ and find the quotient. Answer: The quotient is $$a+6$$.

Step 1: Identify the dividend and divisor

Here, the given polynomial is the dividend, and the linear term is the divisor. So, the dividend is $$a^2 + 9a + 18$$ and the divisor is $$a+3$$.

Step 2: Write down the long division

Now let's set up the problem using long division notation. Write the dividend under the division symbol and the divisor to the left of it: ``` ______ a+3 | a^2 + 9a + 18 ```

Step 3: Divide the first terms

Divide the first term of the dividend, $$a^2$$, by the first term of the divisor, $$a$$. This gives us $$a$$, which will be the first term of the quotient. ``` a ______ a+3 | a^2 + 9a + 18 ```

Step 4: Multiply the divisor by the result

Multiply the divisor, $$a + 3$$, by the term from Step 3, $$a$$. This gives $$a(a + 3) = a^2 + 3a$$. Write this result under the corresponding terms of the dividend and draw a line. ``` a ______ a+3 | a^2 + 9a + 18 - (a^2 + 3a) _________ ```

Step 5: Subtract and bring down the next term

Subtract the result from Step 4 from the first two terms of the dividend. The result is $$9a - 3a = 6a$$. Now bring down the next term, which is $$18$$. ``` a ______ a+3 | a^2 + 9a + 18 - (a^2 + 3a) __________ 6a + 18 ```

Step 6: Divide the remaining terms

Divide the first term of the remaining dividend, $$6a$$, by the first term of the divisor, $$a$$. This gives us $$6$$, which will be the second term of the quotient. ``` a + 6 ______ a+3 | a^2 + 9a + 18 - (a^2 + 3a) __________ 6a + 18 ```

Step 7: Multiply the divisor by the result

Multiply the divisor, $$a + 3$$, by the term from Step 6, $$6$$, and write the result under the remaining dividend. ``` a + 6 ______ a+3 | a^2 + 9a + 18 - (a^2 + 3a) __________ 6a + 18 - (6a + 18) __________ ```

Step 8: Subtract the result

Subtract the result from Step 7 from the remaining dividend: ``` a + 6 ______ a+3 | a^2 + 9a + 18 - (a^2 + 3a) __________ 6a + 18 - (6a + 18) __________ 0 ``` So the result of the polynomial division is $$a + 6$$ with no remainder.

Key Concepts

Long Division of Polynomials

Polynomial division, particularly long division, is a method used to divide a polynomial (the dividend) by another polynomial (the divisor), which is similar in process to dividing numbers. Just like in numerical long division, the goal is to determine how many times the divisor fits into the dividend, in terms of polynomials. This process continues until you either get zero or a remainder that is of a lesser degree than the divisor.

For instance, the exercise requires dividing the polynomial \(a^2 + 9a + 18\) by a linear term \(a + 3\). The process starts with dividing the highest degree term in the dividend by the highest degree term in the divisor to get the first term of the quotient. You then multiply the divisor by this term and subtract it from the dividend, bringing down the next term to repeat the process. This sequence of dividing, multiplying, and subtracting continues until all terms have been handled. Doing so ensures that students can find the quotient step by step, making it easier to understand and verify their work.

Dividend and Divisor in Algebra

In algebra, the terms 'dividend' and 'divisor' play crucial roles in division. The dividend is the polynomial that you want to divide up, and the divisor is the polynomial you are dividing by. It is vital to identify these correctly to perform the division properly. In the given exercise, the dividend is the quadratic polynomial \(a^2 + 9a + 18\), while the divisor is the linear term \(a + 3\).

Understanding the relationship between the dividend and divisor is fundamental because it sets the stage for the entire division process. A common error that students need to avoid is mixing up these two terms, as this can lead to incorrect results. By clearly identifying the dividend and divisor before starting the division process, it's easier for students to follow the proper steps and achieve the correct quotient and remainder, if any.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem is a statement about the remainder when a polynomial is divided by a linear divisor of the form \(x - c\). According to the theorem, if a polynomial \(f(x)\) is divided by \(x - c\), the remainder is \(f(c)\).

For example, in the provided exercise, since the polynomial \(a^2 + 9a + 18\) was divided by \(a + 3\) and we found no remainder, it suggests according to the theorem that \(f(-3) = 0\). Moreover, if we had found a remainder, that remainder would be the value of the polynomial for \(a = -3\). This demonstrates a direct application of the Polynomial Remainder Theorem in confirming the solution of a polynomial division. Students can use this theorem as a quick way to check their work, since any discrepancy in the remainder of a division compared to the value of the dividend at \(c\) points to a mistake in the division process.