Problem 25 - Basic Technical Mathematics with Calculus

Question

Perform the indicated divisions by synthetic division. $$\left(x^{3}+2 x^{2}-3 x+4\right) \div(x+4)$$

Step-by-Step Solution

Answer

The quotient is \(x^2 - 2x + 5\) with a remainder of \(-16\).

Step 1: Set up Synthetic Division

To perform synthetic division for the polynomial \(x^3 + 2x^2 - 3x + 4\) divided by \(x+4\), first identify the root of \(x+4\) which is \(-4\). Write down the coefficients of the polynomial: \(1, 2, -3, 4\). Set up the synthetic division by writing \(-4\) on the left and the coefficients in a row next to it.

Step 2: Begin the Division Process

Bring down the leading coefficient, \(1\), to the bottom row. This is the first step in creating the quotient.

Step 3: Multiply and Add

Multiply \(-4\) by the number just brought down (\(1\)) and write the result below the second coefficient (\(2\)). The result is \(-4\). Add \(2\) (the second coefficient) and \(-4\) together to get \(-2\). Write this sum below.

Step 4: Repeat Multiply and Add

Multiply \(-4\) by the result from the last addition \((-2)\) and write the result (\(8\)) below the next coefficient (\(-3\)). Add \(-3\) and \(8\) to get \(5\), and write this sum below.

Step 5: Final Multiply and Add

Multiply \(-4\) by the result from the last addition \(5\). Write \(-20\) below the last coefficient (\(4\)). Add \(4\) and \(-20\) to get \(-16\). This value represents the remainder of the division.

Step 6: Construct the Quotient

The bottom row (excluding the remainder) gives the coefficients of the quotient polynomial. Thus, the quotient of \(x^3+2x^2-3x+4\) divided by \(x+4\) is \(x^2 - 2x + 5\) with a remainder of \(-16\).

Key Concepts

Polynomial Division

Polynomial division is a method we use to divide one polynomial by another. This technique is much like the long division you learned with numbers but applying it to algebraic expressions. It's valuable because dividing polynomials helps simplify equations and solve complex problems.

In polynomial division, your dividend is the polynomial you're dividing, and your divisor is what you're dividing it by. For example, in the division of \(x^3 + 2x^2 - 3x + 4\) by \(x + 4\), \(x^3 + 2x^2 - 3x + 4\) is the dividend and \(x + 4\) is the divisor.

Using synthetic division is one of the most efficient ways to perform polynomial division when dividing by a linear term, like \(x + 4\). Synthetic division simplifies calculations, especially for larger polynomials, making it accessible even for more complex equations.

Roots of Polynomials

The roots of polynomials are essential in both algebra and calculus because they represent the values that make the polynomial equal to zero.

When dividing by a polynomial or a binomial like \(x + 4\), you start by identifying its root. In our example, the divisor is \(x + 4\), and the root is found by setting it equal to zero: \(x + 4 = 0\), which simplifies to \(x = -4\). This root is the key to setting up synthetic division.

Understanding roots can help when graphing or solving polynomial equations. Roots can indicate where the polynomial crosses the x-axis and can help predict the behavior of the polynomial across different intervals.

Dividing Polynomials

Dividing polynomials, especially through synthetic division, is a streamlined process that reduces the difficulties involved in dividing with long polynomial terms.

The primary steps involved in synthetic division include:

• Identify the root of the divisor, like \(-4\) in our case.
• Write down the coefficients of the dividend polynomial.
• Perform the division process, which involves bringing down, multiplying, and adding coefficients iteratively.
• Construct the final polynomial quotient, making sure to note any remainder.

This method stands out due to its efficiency in calculating polynomial division, especially when dealing with a divisor of the form \(x - c\). By using synthetic division, you gain speed without losing accuracy.

Remainder Theorem

The remainder theorem is a useful tool when working with polynomial division, as it gives direct information about the remainder of a division. If you divide a polynomial \(f(x)\) by a linear divisor \(x - c\), the remainder of this division will be \(f(c)\).

For our polynomial \(x^3 + 2x^2 - 3x + 4\) divided by \(x + 4\), applying synthetic division yields a remainder of \(-16\). Using the remainder theorem, if you substitute \(-4\) into the polynomial, you should end with \(-16\) – this confirms your calculation.

The remainder theorem simplifies the process of checking your work and understanding the relationship between the polynomial and its roots or factors. It's a powerful verification tool and aids in solving polynomial equations effectively.