Problem 28 - Precalculus with Calculus Previews

Question

Use synthetic division to find the quotient \(q(x)\) and remainder \(r\) when \(f(x)\) is divided by the given linear polynomial. $$ f(x)=4 x^{4}+3 x^{3}-x^{2}-5 x-6 ; x+3 $$

Step-by-Step Solution

Answer

The quotient is \( q(x) = 4x^3 - 6x^2 + 17x - 56 \), and the remainder is 162.

Step 1: Identify Coefficients and Setup Division

First, identify the coefficients of the polynomial \( f(x) = 4x^4 + 3x^3 - x^2 - 5x - 6 \). The coefficients are \( 4, 3, -1, -5, -6 \). The linear polynomial is \( x + 3 \), which means we will use \( -3 \) for the synthetic division process.

Step 2: Perform Synthetic Division

Set up the synthetic division by writing \( -3 \) to the left and the coefficients \( 4, 3, -1, -5, -6 \) in a row. Bring down the \( 4 \). Multiply \( 4 \) by \( -3 \) and add it to the next coefficient \( 3 \), getting \( -9 + 3 = -6 \). Continue this process: \(-3 \cdot -6 = 18; -1 + 18 = 17\), then \(-3 \cdot 17 = -51; -5 - 51 = -56\), and finally \(-3 \cdot -56 = 168; -6 + 168 = 162\).

Step 3: Interpret the Result

The results of the synthetic division is the sequence \( 4, -6, 17, -56, 162 \). The quotient \( q(x) \) is formed by the terms \( 4x^3 - 6x^2 + 17x - 56 \), and the remainder \( r \) is \( 162 \).

Step 4: Conclusion

The quotient of the division of \( f(x) \) by \( x + 3 \) is the polynomial \( q(x) = 4x^3 - 6x^2 + 17x - 56 \) and the remainder is \( 162 \). Thus, the division can be expressed as \( f(x) = (x+3)q(x) + r \).

Key Concepts

Polynomial Division

Synthetic division is a simplified form of polynomial division, particularly when dividing by linear polynomials. It's faster and easier than the typical long division method you might use for numbers. This is because it focuses only on the coefficients of the polynomials, which can significantly reduce the complexity.
It involves dividing a polynomial by another polynomial of lesser or equal degree. For example, if you have a polynomial like \( f(x) = 4x^4 + 3x^3 - x^2 - 5x - 6 \) divided by the linear polynomial \( x + 3 \), you would typically seek both a quotient \( q(x) \) and possibly a remainder \( r \).
The process starts by setting up the division using only the coefficients of \( f(x) \) and transforming the divisor \( x+3 \) to use its zero, which is \( -3 \) in the synthetic division setup. This results in a streamlined series of multiplications and additions, leading you directly to the quotient and remainder.

Remainder Theorem

The Remainder Theorem is a fundamental concept in algebra that helps us understand the relationship between division and the remainder. It states that if a polynomial \( f(x) \) is divided by a linear divisor \( x-a \), the remainder of this division is actually \( f(a) \).
This theorem simplifies the process of finding remainders. Instead of performing full polynomial division, you can substitute \( a \) into \( f(x) \) to directly compute the remainder. In synthetic division, the very last number you obtain after processing all the polynomial's coefficients is this remainder.
For example, when dividing \( f(x) = 4x^4 + 3x^3 - x^2 - 5x - 6 \) by \( x+3 \), the remainder is \( 162 \). This value confirms that \( f(-3) = 162 \), showing the power of the Remainder Theorem in synthetic division.

Algebraic Expressions

Algebraic expressions can be intimidating at first, with their mix of numbers, variables, and exponents. However, they follow quite logical patterns once you understand their structure. A polynomial is a type of algebraic expression involving terms with variables raised to whole number exponents and constant coefficients.
When working with polynomials, especially in operations like synthetic division, comprehending their composition is crucial. Each term of a polynomial has a coefficient (the numeric part) and a degree (the exponent of the variable).

• For instance, in \( 4x^4 \), '4' is the coefficient, and '4' is the degree.
• Recognizing these elements allows for effective manipulation, division, and application of the Remainder Theorem.

Understanding algebraic expressions establishes the foundation for simplifying, expanding, and dividing polynomials, leading to efficient problem-solving strategies in algebra.