Problem 9 - College Algebra

Question

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{2 x^{3}+7 x^{2}+9 x-20}{x+3}$$

Step-by-Step Solution

Answer

Quotient, \(q(x)\), is \(2x^{2}+x+6\), and the remainder, \(r(x)\), is \(-38\).

Step 1: Set up the Division

Set up the long division with \(2x^{3}+7x^{2}+9x-20\) divided by \(x+3\).

Step 2: Division

Divide the first term of the dividend, \(2x^{3}\), by the first term of divisor, \(x\). It gives \(2x^{2}\), which is the first term of the quotient. Write this value on top of the division line above the dividend.

Step 3: Multiplication

Multiply \(2x^{2}\) with each term in the divisor \(x+3\). This gives \(2x^{3}+6x^{2}\). Write this below the dividend.

Step 4: Subtraction

Now subtract \(2x^{3}+6x^{2}\) from \(2x^{3}+7x^{2}\) to get the new dividend. It's important to subtract all terms, not just the leading ones. This gives \(x^{2}+9x-20\).

Step 5: Repeat Division

Repeat the division of the first term in the new dividend by the first term of the divisor. This adds \(x\) to the quotient.

Step 6: Repeat Multiplication and Subtraction

Again, multiply the divisor \(x+3\) by the new term in the quotient, \(x\), resulting in \(x^{2}+3x\). Subtract this from the new dividend \(x^{2}+9x-20\) which leads to \(6x-20\).

Step 7: Repeat Division

Now, divide the first term of the new dividend \(6x\) by the first term of the divisor \(x\) to get \(6\) and add it to the quotient.

Step 8: Repeat Multiplication and Subtraction

\Multiply the divisor \(x+3\) with the new term in the quotient, \(6\), which results in \(6x+18\). Subtracting this from the new dividend gives the final remainder, \(r(x)\), as \(-38\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide two polynomials, similar to how long division is used with numbers. When you have a polynomial (like our dividend, \(2x^3 + 7x^2 + 9x - 20\)) that you need to divide by another polynomial (our divisor, \(x + 3\)), you can think of it just like dividing a larger number by a smaller one. Instead of numbers, you're dealing with terms that have variables raised to powers. The process involves:

• Aligning terms by their degree (the power of \(x\)) so you can easily subtract them after multiplication.
• Performing repeated cycles of division, multiplication, and subtraction until there are no more terms to bring down from the dividend or until you reach a remainder that's simpler to handle.

It is important to remember that this method requires careful alignment and attention to signs when performing the subtraction.

Quotient and Remainder

In polynomial division, much like numerical division, the result consists of two parts: the quotient and the remainder. The quotient, \(q(x)\), is the result obtained after dividing the dividend by the divisor. It's the polynomial you get on top of the division line.The remainder, \(r(x)\), is what you have left after you've divided as much as possible. It's similar to the 'leftovers' you might have when dividing numbers, but here, it's another polynomial. The remainder will often be a polynomial of lower degree than the divisor. For example, after using long division on the given polynomials, our quotient came out to \(2x^2 + x + 6\) while the remainder was \(-38\). This means that:\[2x^3 + 7x^2 + 9x - 20 = (x+3)(2x^2 + x + 6) - 38\] Understanding how to find these components is crucial when working with polynomial division.

Algebraic Expressions

Algebraic expressions form the backbone of polynomial division. An algebraic expression is a combination of variables, numbers, and operators (like + or -) that represent a value. Polynomials are a specific type of algebraic expression consisting of multiple terms, each made up of a coefficient and a variable raised to a non-negative integer exponent.In our division exercise, we have the dividend \(2x^3 + 7x^2 + 9x - 20\) and the divisor \(x+3\). Here, each term has its own significance:

• \(2x^3\): The leading term of the polynomial, with a degree of 3.
• \(x+3\): The divisor with a linear term and a constant.

Understanding the structure of terms inside an algebraic expression will help you accurately perform the steps needed in polynomial division. Correctly identifying these terms is key to setting up the problem correctly.

Synthetic Division

Synthetic division is a simplified method used particularly when the divisor is a linear binomial, like \(x + c\). It uses less writing and fewer calculations compared to long division and is typically faster.This method involves:

• Using the coefficients of the dividend separately and performing calculations based on a pattern.
• Only involving the constant term from the divisor, as it is rearranged into a simpler form.

Unfortunately, synthetic division is not applicable in every scenario, working best when dividing by simple divisors. In our scenario, while synthetic division could potentially simplify calculations if \(x+3\) satisfies its criteria, the step-by-step explanation focused on traditional long division for completeness. However, once mastered, synthetic division is a tool that can save time and effort where applicable.