Problem 1 - Algebra and Trigonometry

Question

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=3 x^{2}+5 x-4, \quad D(x)=x+3\)

Step-by-Step Solution

Answer

The polynomial is expressed as \(P(x) = (x + 3)(3x - 4) + 8\).

Step 1: Set Up Synthetic Division

First, identify the divisor's zero by setting the divisor equal to zero: \(x+3=0\). Thus, \(x = -3\). This value, \(-3\), will be used in synthetic division. Write the coefficients of \(P(x) = 3x^2 + 5x - 4\), which are \(3, 5,\) and \(-4\). Prepare a synthetic division layout with these coefficients.

Step 2: Perform Synthetic Division

Bring down the leading coefficient \(3\) as it is. Multiply \(-3\) (from the divisor zero) by \(3\) and write the result under the next coefficient \(5\). Add this result to \(5\), giving a new result of \(5 + (-9) = -4\). Continue by multiplying \(-3\) by the newly calculated \(-4\), and add this to the last coefficient \(-4\) to get the remainder: \(-4 + 12 = 8\). Thus, the quotient \(Q(x)\) is \(3x - 4\) and remainder \(R(x)\) is \(8\).

Step 3: Express the Polynomial Equation

Using the results from synthetic division, express \(P(x)\) in the form of \(D(x) \cdot Q(x) + R(x)\). Substituting the calculated terms gives: \(P(x) = (x + 3)(3x - 4) + 8\). This representation shows \(P(x)\) as a product of the divisor and quotient plus the remainder.

Key Concepts

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form \(x - c\), where the standard division process is streamlined into a concise calculation. This method is particularly useful for its simplicity and efficiency in finding solutions to polynomial equations.

The key steps in synthetic division involve:

• Identifying the zero of the divisor \(D(x)\). If \(D(x) = x + 3\), set \(x + 3 = 0\), which gives \(x = -3\).
• Arranging the coefficients of the dividend polynomial \(P(x)\) while ensuring all terms are in descending degree order including any missing terms.
• Using the zero found to perform synthetic substitution within the coefficients from left to right to calculate the new terms of the quotient \(Q(x)\).

This structured approach not only simplifies the calculation but also offers a quick means of determining the quotient and the remainder for polynomial division.

Polynomial Equations

Polynomial equations are expressions composed of variables and coefficients, where the variables can have non-negative integer powers. They are typically classified based on their degree—the highest power of the variable present in the expression.

For example, given a polynomial equation \(P(x) = 3x^2 + 5x - 4\), it is identified as a quadratic equation due to its highest power being 2. In the context of polynomial division, the aim is to express the polynomial \(P(x)\) as a combination of the divisor \(D(x)\), a quotient \(Q(x)\), and a remainder \(R(x)\).

• The polynomial is divided by a divisor, resulting in a quotient that is a polynomial of one degree less than the original polynomial.
• The process results in an equation of the form: \(P(x) = D(x) \cdot Q(x) + R(x)\). This highlights how polynomial division allows for decomposition and simplification of complex polynomial expressions.

By understanding polynomial equations, one can gain insight into their structure, solutions, and applications across various mathematical problems.

Remainder Theorem

The remainder theorem is a fundamental concept in algebra that provides a quick way to determine the remainder when a polynomial is divided by a linear divisor \(x - c\). According to this theorem, if a polynomial \(P(x)\) is divided by \(x - c\), the remainder of this division is simply \(P(c)\).

In our example, when \(P(x) = 3x^2 + 5x - 4\) is divided by \(x + 3\):

• We first convert \(x + 3\) into \(x - (-3)\), which means the value of \(c\) here is \(-3\).
• Applying the remainder theorem, the remainder \(R\) when divided is equal to \(P(-3)\).

Thus, confirming the synthetic division result of 8, as substituting \(-3\) into \(P(x)\) gives the remainder directly.
This theorem is powerful because it allows you to evaluate polynomials quickly and check the validity of the division result without performing the entire synthetic or long division process again.