Problem 63 - Algebra 1: Concepts and Skills

Question

Find the quotient. Divide \(\left(9 a^{2}-27 a-36\right)\) by \((a+1)\)

Step-by-Step Solution

Answer

So, the quotient of \(\left(9 a^{2}-27 a-36\right)\) divided by \((a+1)\) is \(9a - 36\).

Step 1: Identifying Division

We are given a polynomial \(9 a^{2}-27 a-36\) and we are asked to divide it by a binomial \(a+1\). To find the quotient, it is recommended to use the long division method.

Step 2: Performing Polynomial Long Division

Set up the long division by writing the dividend \(9 a^{2}-27 a-36\) inside the division symbol and the divisor \(a+1\) outside. The first term in the dividend is divided by the first term of the divisor, giving \(9a\). Multiply the divisor by \(9a\) and subtract this from the dividend to find the remainder.

Step 3: Completing Polynomial Long Division

Now bring down the next term \(-36\) from the dividend. Divide \(-36\) by the divisor \(a + 1\), which gives \(-36\). Multiply the divisor by \(-36\) and subtract this from the remainder to find the new remainder, which is zero. This means that the division is complete, and the quotient is \(9a - 36\).

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division of numbers, but it's used to divide one polynomial by another. Just like numerical division, the objective is to determine how many times the divisor fits into the dividend. In polynomials, the terms are arranged in decreasing order of their degree, and we work our way from the highest degree to the lowest, ensuring each term of the dividend is accounted for.

This process provides us with two results: a quotient, which is the result of the division, and often a remainder, which is what's left over when the division cannot be done exactly. In cases where the remainder is zero, as in our example exercise, the divisor is a factor of the dividend. Understanding polynomial division is essential for factoring polynomials, simplifying algebraic fractions, and solving polynomial equations.

Dividing Polynomials Step by Step

When we divide polynomials step by step, we follow a structured approach similar to that of numeric long division. Let's review this process in the context of the given exercise:

• Arrange the polynomials in descending power order.
• Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and write the resulting polynomial below the dividend.
• Subtract this polynomial from the dividend to find the remainder.
• Bring down the next term from the dividend and repeat steps 2 to 4 until all terms are brought down.

If you have a remainder that cannot be divided further by the divisor, the division is complete. The quotient, possibly followed by the remainder over the original divisor, represents the result of the polynomial division.

Algebraic Long Division

Algebraic long division is very similar in technique to the long division we learn with numbers, yet it requires careful attention to variables and their exponents. This process involves writing the dividend inside a division bracket and the divisor on the outside, as well as considering the degree of polynomials.

When performing algebraic long division, we divide term by term, starting from the highest degree on both the dividend and the divisor. As we divide, we multiply, subtract, and bring down the next term systematically. We repeat this process until all the terms of the dividend have been addressed. It's vital to align corresponding terms by degree when subtracting polynomials; otherwise, errors can easily be made. This meticulous effort continues until we get to a stage where no further division is possible, and at this point, if a non-zero remainder exists, it cannot be divided by the divisor.

Quotient of Polynomials

The quotient of polynomials is the result obtained from the division of one polynomial by another, excluding any remainder. In the context of our division exercise, the quotient is the polynomial obtained before we might no longer be left with a dividend to work with or when the remainder becomes zero.

A key thing to remember is that the degree of the quotient will always be less than the degree of the dividend, assuming the divisor is not a constant. In our example, the quotient is a linear polynomial, as the original dividend was quadratic and the divisor was linear. The understanding and simplification of the quotient of polynomials are crucial skills in algebra since they enable us to simplify expressions, and solve equations, and can also be used in graphing polynomial functions.