Problem 13 - College Algebra

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(2 x^{3}+3 x^{2}-4 x+15\right) \div(x+3) $$

Step-by-Step Solution

Answer

The quotient is \(2x^2 - 3x + 5\) and the remainder is 0.

Step 1: Set up the division

Write the polynomial division in long division format. The dividend is \(2x^3 + 3x^2 - 4x + 15\) and the divisor is \((x + 3)\).

Step 2: Divide the leading terms

Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\). This results in \(2x^2\). This will be the first term of the quotient.

Step 3: Multiply and subtract

Multiply \(2x^2\) by \(x + 3\), which gives \(2x^3 + 6x^2\). Subtract this from the original dividend \((2x^3 + 3x^2 - 4x + 15)\), resulting in \(-3x^2 - 4x + 15\).

Step 4: Repeat the process

Now repeat the process using \(-3x^2\). Divide \(-3x^2\) by \(x\) to get \(-3x\). Multiply \(-3x\) by \(x + 3\) and subtract: \(-3x^2 - 9x\), resulting in \(5x + 15\).

Step 5: Final division

Take \(5x\) and divide by \(x\) to get \(5\). Multiply \(5\) by \(x + 3\) which gives \(5x + 15\), and finally, subtract \(5x + 15\) from \(5x + 15\), this results in 0.

Step 6: Conclusion

The final quotient is \(2x^2 - 3x + 5\) with a remainder of 0.

Key Concepts

Polynomial Division

Long division in the context of polynomials is akin to long division with numbers but tailored for algebraic expressions. It is a method used when dividing one polynomial by another of a lesser degree.
Here, you begin by arranging the dividend and divisor. The dividend is the polynomial you're dividing, while the divisor is the one you're dividing by. In our example:

• The dividend is \(2x^3 + 3x^2 - 4x + 15\)
• The divisor is \(x + 3\)

Then, just like arithmetic division, you divide the leading term of the dividend by the leading term of the divisor. This results in the first term of your quotient. You multiply the entire divisor by this term, subtract it from the original dividend, and repeat these steps with the new expression formed. Continue this until there are no terms left in the dividend that can be divided by the divisor. This systematic approach helps simplify complex polynomial expressions.

Quotient and Remainder

When using long division to divide polynomials, the ultimate goal is to determine both the quotient and the remainder. The quotient is the result of the division without considering the divisor, while the remainder is what's left after the division process is unable to continue.
In the example provided, the division process systematically breaks down the dividend \(2x^3 + 3x^2 - 4x + 15\) into a simpler form. Step by step, each term of the quotient is determined until no further division can occur beyond a constant.

• Quotient: \(2x^2 - 3x + 5\)
• Remainder: 0

The remainder is zero, indicating that our divisor \(x + 3\) precisely divides the dividend without leaving any leftover terms. Understanding quotient and remainder gives students insight into how polynomials relate and interact with one another.

Algebraic Expressions

Algebraic expressions are mathematical phrases composed of numbers, operators, and variables. They form the backbone of algebra and can represent a wide variety of values depending on the substitutions for the variable.
In polynomial division, we work specifically with polynomial expressions. A polynomial is a type of algebraic expression involving sums of powers of one or more variables multiplied by coefficients.

• The dividend in this exercise, \(2x^3 + 3x^2 - 4x + 15\), is a polynomial of degree three.
• The divisor \(x + 3\) is of degree one.

A solid grasp of algebraic expressions allows students to manipulate and simplify expressions more readily, providing foundational skills for solving more complex algebraic equations. Understanding how polynomials are structured and behave during division enhances problem-solving abilities and builds a deeper comprehension of algebraic principles.