Problem 83 - Intermediate Algebra

Question

Let \(f(x)=4 x^{4}+20 x^{3}-x^{2}-2 x+15\) and \(g(x)=x+5\) Find \(\frac{f(x)}{g(x)}\) in simplified form.

Step-by-Step Solution

Answer

\( \frac{f(x)}{g(x)} = 4x^3 - x + 3 \)

Step 1: Understand the Problem

We are given two functions, \( f(x) = 4x^4 + 20x^3 - x^2 - 2x + 15 \) and \( g(x) = x + 5 \). We need to simplify the expression \( \frac{f(x)}{g(x)} \). This simplification usually involves polynomial division.

Step 2: Set Up Polynomial Long Division

To divide \( f(x) \) by \( g(x) \), we start by writing \( f(x) \) as the dividend and \( g(x) \) as the divisor. Polynomial long division is similar to numerical long division.

Step 3: Divide the Leading Terms

Divide the leading term of the dividend \( 4x^4 \) by the leading term of the divisor \( x \) to get \( 4x^3 \). This becomes the first term of the quotient.

Step 4: Multiply and Subtract

Multiply \( 4x^3 \) by \( g(x) \), giving \( 4x^4 + 20x^3 \), and subtract this from \( f(x) \). The new dividend is \( -x^2 - 2x + 15 \).

Step 5: Repeat the Division Process

Continue the process with the new dividend. Divide \( -x^2 \) by \( x \) to get \( -x \), multiply to get \( -x^2 - 5x \), subtract to find \( 3x + 15 \).

Step 6: Divide the Remaining Terms

Divide \( 3x \) by \( x \) to get \( 3 \). Multiply \( 3 \) by \( g(x) \) to obtain \( 3x + 15 \), subtract it from \( 3x + 15 \) to find the remainder 0.

Step 7: Compile the Quotient

After the divisions, we compile the terms obtained through each step. In this case, the final quotient is \( 4x^3 - x + 3 \) with a remainder of 0.

Step 8: Write the Simplified Form

Since the remainder is 0, \( \frac{f(x)}{g(x)} = 4x^3 - x + 3 \). Thus, the simplified form of \( \frac{f(x)}{g(x)} \) is complete.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of variables, each multiplied by a coefficient. These functions appear very frequently in various areas of mathematics and science.

Here's what makes a polynomial function special:

• They consist of terms with variables raised to non-negative integer exponents.
• The coefficients are real numbers, which may be whole numbers, fractions, or decimals.
• The degree of a polynomial is determined by the highest power of the variable in the function.

In the given exercise, the polynomial function is

\( f(x) = 4x^4 + 20x^3 - x^2 - 2x + 15 \)

It has five terms and the degree is 4, due to the term \(4x^4\). Understanding the structure of polynomial functions is crucial when performing operations like addition, subtraction, and especially division.

Quotient and Remainder

When dividing polynomials, much like numbers, we are often interested in the quotient and remainder. This follows the division algorithm principle:

Any dividend \(f(x)\) can be expressed in terms of the divisor \(g(x)\) in the following way:

\[ f(x) = g(x) \, \cdot \, q(x) + r(x) \]
Here, \(q(x)\) is the quotient and \(r(x)\) is the remainder.

Key points:

• The degree of the remainder \(r(x)\) is always less than the degree of \(g(x)\).
• If \(r(x) = 0\), as in this exercise, the division is exact, indicating that \(g(x)\) is a factor of \(f(x)\).

In our exercise, we divided \( f(x) \) by \( g(x) = x + 5 \) and found that the quotient is \( 4x^3 - x + 3 \) with no remainder. This means that after division, \( f(x) \) is exactly equal to \( (x + 5) \, \cdot \, (4x^3 - x + 3) \). This is a helpful way to interpret polynomial division results!

Polynomial Simplification

Simplification of polynomials often involves operations such as division, much like what we've done in the exercise, to reduce the expression into a form that is simpler to work with. Simplifying involves finding the simplest equivalent expression that contains the same information.

In our example:

• We performed polynomial long division on \( f(x) = 4x^4 + 20x^3 - x^2 - 2x + 15 \) using \( g(x) = x + 5 \) as the divisor.
• Through division, we found that \( f(x) \) simplifies to \( 4x^3 - x + 3 \) with no remainder.
• This simplified form is more concise and easier to handle in further calculations or evaluations.

Having the simplified form \( 4x^3 - x + 3 \) makes it easier to analyze or plug into additional mathematical operations. Simplification saves time and reduces potential errors in computations or modeling situations where polynomial expressions are involved.