Problem 5 - College Algebra

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(3 x^{2}+23 x+14\right) \div(x+7) $$

Step-by-Step Solution

Answer

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

Step 1: Set up the Division

Write the dividend, \(3x^2 + 23x + 14\), and the divisor, \(x+7\), in long division format, placing \(3x^2 + 23x + 14\) inside the division bracket and \(x+7\) outside.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \(3x^2\), by the leading term of the divisor, \(x\). This gives \(3x\). Write \(3x\) as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply \(3x\) by \(x+7\), giving \(3x^2 + 21x\). Subtract \(3x^2 + 21x\) from \(3x^2 + 23x + 14\), resulting in a remainder of \(2x + 14\).

Step 4: Repeat Division

Divide the leading term of the new remainder, \(2x\), by the leading term of the divisor, \(x\). This gives \(2\). Write \(2\) as the next term of the quotient.

Step 5: Multiply and Subtract Again

Multiply \(2\) by \(x+7\), yielding \(2x + 14\). Subtract \(2x + 14\) from \(2x + 14\), resulting in a remainder of \(0\).

Step 6: State the Quotient and Remainder

The quotient is \(3x + 2\) and the remainder is \(0\). This completes the long division process.

Key Concepts

Polynomial Division

Polynomial division is an essential method, similar to long division with numbers, used to divide one polynomial by another. This technique helps us in simplifying algebraic expressions, breaking them down into more manageable parts.
It's especially useful when handling complex polynomials that require simplification or solving.
In our exercise, the polynomial division was applied to divide \(3x^2 + 23x + 14\) by \(x + 7\). The process involves dividing each term of the polynomial, starting with the term with the highest degree. This systematic approach ensures accuracy and helps us find both the quotient and any remainder resulting from the division.
The steps include setup, dividing leading terms, multiplying, subtracting, and repeating until no remainder remains, or a zero remainder is found. Understanding these steps is critical to mastering polynomial division, which is a powerful tool in algebra.

Quotient and Remainder

In algebra, just like with numbers, when dividing two polynomials, we get a quotient and sometimes a remainder.
The quotient is the result of division, while the remainder is what's left after the division. When using the long division method, you continue the process until the degree of the remainder is less than the degree of the divisor.
For instance, in our worked example, the division led us to a quotient of \(3x + 2\) with a remainder of \(0\). This indicates that the division was exact and there was no leftover term. The expression was perfectly divisible.
Understanding the quotient and remainder helps verify algebraic division, allowing one to check their work by multiplying the quotient by the divisor and adding the remainder to see if it matches the original dividend (\(3x^2 + 23x + 14\)). It's a useful method to confirm the validity of the division performed.

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants formed through operations such as addition, subtraction, and multiplication. These expressions are fundamental to algebra and are used to convey mathematical ideas efficiently.
In the division problem, we dealt with an algebraic expression \(3x^2 + 23x + 14\) divided by another expression, \(x + 7\). Knowing how to manage these expressions through division simplifies solving algebraic equations and finding roots, simplify fractions, and more.
Moreover, the ability to manipulate algebraic expressions through division helps in understanding deeper concepts like factoring, simplifying complex fractions, and solving polynomial equations. Mastery of these expressions ensures a solid foundation in algebra and prepares students for more advanced mathematical topics, making this knowledge highly valuable.