Problem 57 - Intermediate Algebra

Question

Perform each division. \(\frac{4 a^{3}+a^{2}-3 a+7}{a+1}\)

Step-by-Step Solution

Answer

The division is \(4a^2 - 3a\) with a remainder of 7.

Step 1: Setup Polynomial Long Division

To divide \(4a^{3} + a^{2} - 3a + 7\) by \(a + 1\), we'll use polynomial long division. We represent the dividend as \(4a^3 + a^2 - 3a + 7\) and the divisor as \(a + 1\).

Step 2: Divide Leading Terms

Divide the leading term of the dividend \(4a^3\) by the leading term of the divisor \(a\). \[\frac{4a^3}{a} = 4a^2\]. This is the first term of our quotient.

Step 3: Multiply and Subtract

Multiply \(4a^2\) by the entire divisor \(a + 1\), which gives \(4a^3 + 4a^2\). Subtract this result from the original dividend: \[ (4a^3 + a^2 - 3a + 7) - (4a^3 + 4a^2) = -3a^2 - 3a + 7\].

Step 4: Repeat Process with New Dividend

Take the new dividend \(-3a^2 - 3a + 7\) and divide the leading term \(-3a^2\) by \(a\). This gives \[-3a\]. Multiply \(-3a\) by \(a + 1\) to get \(-3a^2 - 3a\). Subtract: \[ (-3a^2 - 3a + 7) - (-3a^2 - 3a) = 7\].

Step 5: Complete the Division

Now with the remaining dividend of \(7\), divide \(7\) by \(a + 1\) using constant terms. Since the degree is less along with any further division being zero, 7 is the remainder. So, our quotient becomes \(4a^2 - 3a + 0\) with a remainder of 7.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another. This method is quite similar to the long division process used with numbers. It is particularly useful when you need a systematic approach to find both the quotient and the remainder of a division involving polynomials.

To begin, write the dividend (the polynomial you're dividing) and the divisor (the polynomial you're dividing by) in descending order of their terms. In our example, the dividend is represented as \(4a^3 + a^2 - 3a + 7\) and the divisor is \(a + 1\). The division process is performed step-by-step, similar to numerical long division but focusing on terms and coefficients. Each step involves dividing, multiplying, and subtracting terms to simplify the original polynomial, eventually yielding a quotient and a remainder.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, while the remainder is what is left over after the division process.

It is important to note that the degree of the remainder should always be less than the degree of the divisor, ensuring that the division is complete. In our division example, the quotient we found was \(4a^2 - 3a\) and the remainder was 7.

The expression of a polynomial division can be written in the form: \( \text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} \).
For our specific case: * Dividend: \(4a^3 + a^2 - 3a + 7\) * Divisor: \(a + 1\) * Quotient: \(4a^2 - 3a\) * Remainder: 7 This means the division of \(4a^3 + a^2 - 3a + 7\) by \(a + 1\) can be rewritten as: \( (a + 1)(4a^2 - 3a) + 7 \).

Division of Polynomials

The division of polynomials is an essential concept in algebra and helps simplify complex expressions and solve polynomial equations. This operation involves breaking down a complex polynomial into simpler parts by dividing it by another polynomial.

During the division process, you repeatedly divide the leading term of the remaining polynomial by the leading term of the divisor until the degree of what remains is less than the degree of the divisor. While performing polynomial division, make sure to align the terms according to their degree and perform accurate arithmetic operations at each stage.

Mastering this technique facilitates understanding polynomial identities and solving higher-level algebraic problems. It also aids in evaluating limits in calculus when dealing with polynomial expressions.

Intermediate Algebra Concepts

Intermediate algebra introduces students to more advanced algebraic concepts, including polynomial division, which can seem complex at first but is crucial for further study in mathematics. These concepts build the foundational skills needed to understand functions, solve equations, and approach calculus problems in the future.

Some of the key elements in understanding intermediate algebra include:

• Recognizing polynomial expressions and their terms.
• Understanding degree and coefficients of polynomials.
• Mastering operations like addition, subtraction, multiplication, and division of polynomials.
• Applying algebraic identities and theorems to simplify and solve problems.

As students master these skills, they become better equipped to tackle complex mathematics, making polynomial division not only a valuable skill in its own right but also a stepping stone to advanced mathematical concepts and problem-solving.