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Magazine Chapter 3: Problem 9
Divide using long division. State the quotient, q(x), and the remainder, r(x). $$\frac{2 x^{3}+7 x^{2}+9 x-20}{x+3}$$
Short Answer
Expert verified
The quotient, q(x), is \(2x^2 + x + 6\) and the remainder, r(x), is \(-38\).
Step by step solution
01
Set up the division.
The first step in long division of polynomials is to set up the division. We take \(x+3\) to be the divisor and \(2 x^{3}+7 x^{2}+9 x-20\) to be the dividend. We start by dividing the highest degree term of the dividend, \(2x^3\), by the highest degree term of the divisor, \(x\). This gives us \(2x^2\), which is the first term of the quotient.
02
Multiply and subtract
Next, we multiply the divisor, \(x+3\) by the first term of the quotient, \(2x^2\), to give \(2x^3 + 6x^2\). We then subtract this from the original dividend to get a new polynomial which is \(x^2+9x-20\). We then divide the highest degree term of this polynomial by the highest degree term of the divisor and add to the quotient.
03
Repeat the process
We continue repeating the process of multiplying and subtracting until we can no longer divide. Doing so, we divide \(x^2\) by \(x\) to get \(x\), subtract \(3x\) from \(9x\) to get \(6x - 20\). Again we divide \(6x\) by \(x\) to get \(6\), subtract \(18\) from \(-20\) to get \(-38\), which is our final remainder as it can no longer be divided by \(x+3\). The quotient then is \(2x^2 + x + 6\).
04
State the quotient and remainder
Having performed the long division fully, we state the quotient, denoted q(x), and the remainder, denoted r(x). The quotient, q(x), is \(2x^2 + x + 6\) and the remainder, r(x), is \(-38\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Long Division
Polynomial long division is similar to long division with numbers, but it involves algebraic expressions. This method is used to divide one polynomial by another, resulting in a quotient and sometimes a remainder.
It's an orderly method where you write the dividend inside the division symbol and the divisor outside. Then, you execute a series of steps to systematically reduce the polynomial. Here鈥檚 a simple guide to follow:
It's an orderly method where you write the dividend inside the division symbol and the divisor outside. Then, you execute a series of steps to systematically reduce the polynomial. Here鈥檚 a simple guide to follow:
- Set up the division as you would in numerical long division.
- Divide the leading term of the dividend by the leading term of the divisor.
- Multiply the entire divisor by the result from the previous step and subtract from the dividend.
- Repeat the process with the new polynomial that forms after subtraction, till the degree of the remainder is less than the degree of the divisor or zero.
Quotient
The quotient in polynomial division is the result you get once you've divided the polynomial as much as possible. In terms of algebraic expressions, it鈥檚 the polynomial that forms part of the result that matches or "fits" into the original polynomial being divided, excluding the smaller remainder that's left over.
In our example using the polynomial division process, the quotient is the expression formed step by step:
In our example using the polynomial division process, the quotient is the expression formed step by step:
- First, divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\) to get \(2x^2\).
- Repeat the multiplication and subtraction to form a new polynomial to continue with till you can't divide anymore.
- For our case, the final quotient is \(2x^2 + x + 6\).
Remainder
The remainder in polynomial long division emerges when no more division can be handled due to a smaller degree compared to the divisor. Just like in regular numerical division, this remainder is what's left over after the division process is complete.
The remainder can sometimes be zero, which indicates that the divisor divides perfectly into the dividend. However, when it's not zero, it's a smaller polynomial.
The remainder can sometimes be zero, which indicates that the divisor divides perfectly into the dividend. However, when it's not zero, it's a smaller polynomial.
- The procedure involves continuously multiplying and subtracting the previous result, until the remaining polynomial鈥檚 degree is lesser than that of the divisor.
- In our problem, after the full division, we are left with \(-38\) as the remainder when dividing \(2x^3 + 7x^2 + 9x - 20\) by \(x + 3\).
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. They form the basis for polynomial division and define the structure of polynomials involved.
An algebraic expression, particularly polynomials, consists of several terms connected by operations. Division of two such polynomials is a common operation facilitated by these structured expressions.
Consider the key terms for understanding algebraic expressions:
An algebraic expression, particularly polynomials, consists of several terms connected by operations. Division of two such polynomials is a common operation facilitated by these structured expressions.
Consider the key terms for understanding algebraic expressions:
- **Coefficients:** Numbers in front of variables (e.g., in \(2x\), 2 is the coefficient).
- **Variables:** Symbols that represent numbers in an expression (e.g., \(x, y\)).
- **Exponents:** They denote the power to which the variable is raised (e.g., in \(x^2\), 2 is the exponent).
- **Terms:** Separate elements of an expression added together (e.g., \(2x^2, 3x, 4\) in the polynomial \(2x^2 + 3x + 4\)).
