Problem 24 - A Graphical Approach to College Algebra

Question

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{4}-3 x^{3}-5 x^{2}+2 x-16 ; \quad x-3$$

Step-by-Step Solution

Answer

The quotient is \(x^3 - 5x - 13\) and the remainder is \(-55\).

Step 1: Set Up the Division

We are tasked with dividing the polynomial \( P(x) = x^4 - 3x^3 - 5x^2 + 2x - 16 \) by \( x - 3 \). We will use synthetic division as it is suitable for dividing by linear factors like \( x - c \). For synthetic division, we arrange the coefficients of \( P(x) \) in order: \([1, -3, -5, 2, -16]\), and we use \( 3 \) from \( x - 3 \) as the number outside the division.

Step 2: Perform Synthetic Division

1. Bring down the leading coefficient, 1.2. Multiply 3 (from \( x-3 \)) by the latest number in the bottom row and add to the next coefficient: - \(1 \times 3 = 3\), \(-3 + 3 = 0\) - \(0 \times 3 = 0\), \(-5 + 0 = -5\) - \(-5 \times 3 = -15\), \(2 - 15 = -13\) - \(-13 \times 3 = -39\), \(-16 - 39 = -55\)3. The numbers at the bottom (except the last) represent the coefficients of the quotient, and the last number is the remainder.

Step 3: Write the Result

The result from the synthetic division gives us \( 1, 0, -5, -13 \) as coefficients of the quotient, and \(-55\) as the remainder. Therefore, the quotient is \( x^3 - 5x - 13 \) and the remainder is \(-55\). This means that:\[P(x) = (x-3)(x^3 - 5x - 13) - 55\].

Key Concepts

Polynomial Division

Polynomial division involves dividing a polynomial by another polynomial, usually of lower degree. The process simplifies complex expressions and helps us find relationships between polynomials.

There are primarily two methods of dividing polynomials:

• Long Division: This method is similar to numerical long division. It works for dividing polynomials by both linear and non-linear divisors.
• Synthetic Division: This is a short-cut method specifically designed for dividing a polynomial by a linear polynomial (of the form \(x-a\)). It’s quicker and involves less writing.

In the given exercise, we used synthetic division since the divisor was the linear polynomial \(x - 3\).

For synthetic division, you only need the coefficients of the polynomials. This method simplifies the division process and leads to efficient calculation of quotient and remainder, which can be crucial for verifying polynomial identities or solving equations.

Remainder Theorem

The Remainder Theorem is a handy tool when working with polynomial division. It states that when a polynomial \(P(x)\) is divided by a linear divisor \(x-c\), the remainder of this division is \(P(c)\).

This theorem is incredibly useful because:

• It allows you to find the value of a polynomial at a specific point without performing the entire division.
• If the remainder is zero, \(c\) is a root of the polynomial \(P(x)\).

In our exercise, after performing synthetic division, we found the remainder to be -55. According to the Remainder Theorem, this means:

• \(P(3) = -55\)

We substituted \(3\) into \(P(x)\) to verify this directly, confirming the remainder.

Linear Factor Division

Dividing by a linear factor such as \(x - c\) is common in algebra for simplifying polynomials or uncovering roots.

When we divide a polynomial \(P(x)\) by a linear factor \(x-c\), the result is a polynomial of one degree less plus a remainder. This is expressed as:

• \(P(x) = (x-c)Q(x) + R\)

Here, \(Q(x)\) is the quotient, and \(R\) is the remainder.

In our example, the quotient was found to be \(x^3 - 5x - 13\), and the remainder was -55.

Understanding linear factor division helps in various algebraic applications, such as solving polynomial equations and factoring higher degree polynomials into simpler, more manageable parts.