Problem 22 - Precalculus

Question

Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{3 x^{2}+4 x-1}{x-1}$$

Step-by-Step Solution

Answer

Quotient is \(3x + 7\) with a remainder of 6.

Step 1: Write Down Polynomial Coefficients

Identify the coefficients of the polynomial \(3x^2 + 4x - 1\). These are 3, 4, and -1.

Step 2: Set Up Synthetic Division

Write the root of the divisor \(x-1=0\), which gives \(x=1\). This is used as the number to divide the coefficients synthetically.

Step 3: Perform Synthetic Division

1. Bring down the first coefficient (3) under the division line.2. Multiply it by 1 (the root) and write the result below the next coefficient (4).3. Add: \(4 + 3 = 7\).4. Multiply 7 by 1 and write the result below the next coefficient (-1).5. Add: \(-1 + 7 = 6\).Thus, the quotient is \(3x + 7\) and the remainder is 6.

Step 4: Write Division Result in Standard Form

The division process gives us \(q(x) = 3x + 7\) and \(R(x) = 6\). Therefore, the polynomial can be expressed as:\[p(x) = d(x) \cdot q(x) + R(x) \]Substituting gives:\[3x^{2} + 4x - 1 = (x - 1)(3x + 7) + 6\].

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to long division with numbers. Imagine it as a way to distribute a larger polynomial over a smaller one to find how many times the smaller one fits into the larger one.
When using synthetic division, we apply this technique but in a more simplified shortcut form. This saves time and reduces errors, especially when dividing by a linear divisor like \(x - 1\).
In the given exercise, the polynomial \(3x^2 + 4x - 1\) is divided by \(x - 1\) to find both the quotient and remainder. Using synthetic division here makes the process efficient. Always ensure to line up your coefficients correctly and perform multiplication and addition carefully.

• Identify the coefficients of the polynomial dividend.
• Use the root of the divisor's factor for synthetic division.
• Follow the synthetic division steps precisely for accuracy.

Remainder Theorem

The Remainder Theorem is a critical concept to understand when working with polynomials. It states that when a polynomial \(p(x)\) is divided by a linear divisor \(x - c\), the remainder of this division is \(p(c)\).
For instance, if dividing \(3x^2 + 4x - 1\) by \(x - 1\), plugging \(x = 1\) into the polynomial should yield the remainder. An essential part of this theorem is its ability to verify the result of division.
In our exercise, we found that the remainder was 6 after synthetic division. To double-check, calculate \(p(1) = 3(1)^2 + 4(1) - 1 = 6\). This confirms the result. The Remainder Theorem acts as a useful tool for quick validation.

• Know the polynomial to substitute into the divisor's root.
• Double-check your synthetic division results using this theorem.
• The remainder tells if the root is a factor (remainder = 0).

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operators (such as +, -, *, /) that form a polynomial. In this context, they allow us to express and manipulate mathematical relationships.
When learning about polynomial and synthetic division, it's essential to understand the structure of algebraic expressions. For example, in \(3x^2 + 4x - 1\), variables and coefficients are organized into terms separated by plus or minus signs.
With the division result \(3x + 7\) and remainder 6, expressing in the form \(p(x) = d(x) \cdot q(x) + R(x)\) helps structure the solution clearly. It signifies that \(3x^2 + 4x - 1 = (x - 1)(3x + 7) + 6\).

• Recognize and understand the roles of variables, coefficients, and constants in expressions.
• Use expressions to represent division outcomes elegantly.
• Being comfortable with expressions aids in simplifying complex polynomial tasks.