Problem 7 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(4 x^{2}+19 x-32\right) \div(x+6) $$

Step-by-Step Solution

Answer

Quotient: \(4x - 5\); Remainder: \(-2\).

Step 1: Write the Polynomial in Standard Form

Ensure that the polynomial is already in standard form. Given: \(4x^2 + 19x - 32\). The polynomial is in decreasing order of the variable power, so it is in standard form.

Step 2: Set Up the Synthetic Division

To begin synthetic division, identify the root of the divisor \((x + 6)\). This root is \(x = -6\). Write this outside the synthetic division bracket. On the same line, list the coefficients of the dividend \(4x^2 + 19x - 32\), which are \(4\), \(19\), and \(-32\).

Step 3: Perform Synthetic Division

1. Bring down the leading coefficient \(4\) to the bottom row.2. Multiply \(4\) by \(-6\) (the root) and write the result \(-24\) underneath the next coefficient \(19\).3. Add \(19 + (-24) = -5\) and write \(-5\) in the bottom row.4. Multiply \(-5\) by \(-6\) to get \(30\), and write \(30\) underneath the last coefficient \(-32\).5. Add \(-32 + 30 = -2\) and write \(-2\) in the remainder position.

Step 4: Interpret the Results

The bottom row now reads \([4, -5, -2]\), where \(4\) is the coefficient of \(x\), \(-5\) is the constant term, and \(-2\) is the remainder. So, the quotient is \(4x - 5\) and the remainder is \(-2\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial of the first or lower degree. It's comparable to the long division method used in arithmetic. In our specific example, synthetic division is a streamlined technique particularly useful when the divisor is a linear binomial of the form \(x - r\).

The process involves:

• Taking only the coefficients of the polynomial dividend.
• Using the root of the divisor (with the opposite sign) as a factor.
• Simplifying calculations by reducing the need for writing variable terms.

By focusing on coefficients and a single number from the divisor, synthetic division quickly provides a quotient and remainder.

Thus, for the polynomial \(4x^2 + 19x - 32\) divided by \(x + 6\), the use of synthetic division simplifies the process as compared to the conventional polynomial division approach.

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra. It states that when a polynomial \(f(x)\) is divided by a linear divisor \(x - r\), the remainder is the same as \(f(r)\), the value of the polynomial evaluated at \(x = r\).

In our example, we use synthetic division to divide \(4x^2 + 19x - 32\) by \(x + 6\). Through this method, the remainder is found to be \(-2\). We can verify this result by plugging \(x = -6\) into the polynomial:
\[4(-6)^2 + 19(-6) - 32 = 144 - 114 - 32 = -2\]
The consistency of this result confirms the utility of the Remainder Theorem in predicting and checking the remainder of polynomial division.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the backbone of algebra, allowing us to construct models of real-world situations and solve equations.

Polynomials are a type of algebraic expression characterized by terms that are sums or differences of monomials. Understanding how to manipulate these, particularly through division, is critical for simplifying expressions and solving equations.

• Coefficients are the numerical parts of terms, crucial in synthetic division.
• Variables (e.g., \(x\)) represent unknown quantities.
• Operations, such as addition and multiplication, dictate the interaction between terms.

Through the exercise of synthetic division, students see how complex algebraic expressions can be broken down into simpler components, facilitating easier computation and understanding.