Problem 306 - Precalculus

Question

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

Quotient: \(3x + 2\); Remainder: \(0\).

Step 1: Setup the Long Division

Write the divisor \(x + 7\) outside the long division symbol and the dividend \(3x^2 + 23x + 14\) under the long division bracket. This setup is similar to traditional numeric long division.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(3x^2\) by the leading term of the divisor \(x\). The result is \(3x\). Write this as the first term of the quotient above the long division bar.

Step 3: Multiply and Subtract

Multiply the entire divisor \(x + 7\) by the first term of the quotient \(3x\) to get \(3x(x + 7) = 3x^2 + 21x\). Write \(3x^2 + 21x\) under the dividend and subtract it from \(3x^2 + 23x\). This results in \((3x^2 + 23x) - (3x^2 + 21x) = 2x\).

Step 4: Bring Down the Next Term

Bring down the next term in the dividend, which is \(+14\), to the right of \(2x\). This results in the new dividend part \(2x + 14\).

Step 5: Repeat the Process

Divide \(2x\) by \(x\), which gives \(2\). Add \(2\) to the quotient. Multiply the divisor \(x + 7\) by \(2\) to get \(2x + 14\).

Step 6: Subtract Again

Subtract \(2x + 14\) from \(2x + 14\). The result is \(0\), meaning there is no remainder. Thus, the division process is complete.

Key Concepts

Polynomial Division

Polynomial division is a method for dividing one polynomial by another, similar to long division with numbers. It helps simplify complex expressions and can reveal useful factors or roots. In this exercise, we used long division to divide the polynomial \(3x^2 + 23x + 14\) by \(x + 7\). The goal is to find out how many times the divisor fits into the dividend. This process reveals the quotient and possibly a remainder.To start, set up the problem with the dividend and divisor ready for division, much like setting up a long division problem in arithmetic. Identify the leading term of both the dividend and divisor to begin simplifying step by step. This helps make sure each step reduces the polynomial efficiently.

Remainder

When dividing polynomials, the remainder is what's left after the division process is complete. Sometimes, the dividend can't completely be divided by the divisor, resulting in a remainder.In the exercise, once we performed the polynomial division of \(3x^2 + 23x + 14\) by \(x + 7\), there was no remainder left. This means the division was exact, and the dividend is completely divisible by the divisor without any leftover terms.If there were a remainder, it would be a polynomial of lesser degree than the divisor and would be added to the final quotient as its remainder.

Quotient

The quotient in polynomial division is the result obtained after the dividend has been divided by the divisor, excluding any remainder. It represents how many times the divisor fits into the dividend.In our example, after dividing \(3x^2 + 23x + 14\) by \(x + 7\), we obtained a quotient of \(3x + 2\). This quotient indicates that \(x + 7\) fits exactly into the dividend \(3x^2 + 23x + 14\), with no remainder.The quotient is an important part of understanding how the two expressions relate to each other. It's expressed as a polynomial with a degree determined by subtracting the degree of the divisor from the degree of the dividend.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations like addition and multiplication. They can represent numbers, designate geometric sizes, or characterize real-world quantities and their relationships.In the context of polynomial division, both the dividend \(3x^2 + 23x + 14\) and the divisor \(x + 7\) are algebraic expressions. The dividend is a quadratic expression (degree 2), and the divisor is a linear expression (degree 1).When dividing polynomial expressions, the long division method operates on these expressions to produce a simplified expression, which is essential in solving more complex algebraic equations or simplifying expressions.