Question

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=4 x^{4}+6 x^{3}+4 x^{2}-5 x+13, \quad k=-\frac{1}{2}$$

Short answer

Upon division, the quotient \(q(x)=4x^3+8x^2+12x+6\) and the remainder \(r=16\) are obtained. Thus, the function can be rewritten as \(f(x)=(x-(-1/2))(4x^3+8x^2+12x+6) + 16\). Then, when \(k=-1/2\) is substituted into \(f(x)\), \(f(-1/2)=16\), which indeed equals \(r\).

Step-by-step solution

Measure Polynomial Division

Use the polynomial long division with \(f(x)\) as the dividend and \((x+1/2)\) as the divisor. This will allow to obtain \(q(x)\) and \(r\).

Complete Polynomial Division

Upon doing the polynomial division, the expected result is \(q(x)=4x^3+8x^2+12x+6\) and \(r=16\).

Rewrite the function

With the obtained results, \(f(x)\) can now be rewritten in the required format: \(f(x)=(x-(-1/2))(4x^3+8x^2+12x+6) + 16\).

Calculate the value of \(f(k)\)

Substitute \(k=-1/2\) into \(f(x)\). Since \(f(k)\) equals the remainder \(r\) when divided by \((x-k)\), \(f(-1/2)=16\).

Key concepts

Remainder Theorem

When diving into the realm of polynomial functions, it's crucial to understand the Remainder Theorem, a fundamental concept in algebra. Imagine you have a polynomial function, like the one in our exercise, and you want to find out what happens when you divide it by a linear factor of the form (x – k). The Remainder Theorem tells us that the remainder of this division is simply the function evaluated at k: (f(k)).

Put differently, if you substitute the value of k into the polynomial function (f(x)), the result will be the same as the remainder when (f(x)) is divided by (x - k). In our case, the polynomial (4x^4 + 6x^3 + 4x^2 - 5x + 13) divided by (x + 1/2) left us with a remainder of 16. According to the Remainder Theorem, (f(-1/2)) should also equal 16, and indeed, when the value of (-1/2) is substituted into the function, it verifies the theorem.

• The Remainder Theorem simplifies the process of evaluating polynomials at specific points.
• It is particularly useful for determining factors of a polynomial and for sketching its graph.
• Understanding this theorem is a stepping stone to mastering polynomial division techniques like synthetic division.

Working with polynomial functions often requires verification and the Remainder Theorem provides a quick and reliable method to ensure the division was performed correctly.

Synthetic Division

Synthetic division is your algebraic ally when tackling polynomial division, especially when dividing by a linear factor like (x - k). It's a streamlined alternative to the more labor-intensive long division, offering a quicker and more efficient way to divide polynomials and find the remainder, if any.

To perform synthetic division, you write down the coefficients of the polynomial and then bring down the first coefficient. After that, you multiply and add according to a set pattern until you have worked through all the coefficients. This method gives you both the quotient polynomial and the remainder without writing out the full division. Considering our exercise, we would see numbers lined up and manipulated, resulting in a quotient of (4x^3 + 8x^2 + 12x + 6) and a remainder of 16 in far less time than traditional long division.

• Synthetic division requires less space and fewer steps than long division, making it preferable for polynomials.
• It is only applicable for divisors of the first degree (like (x - k)).
• This approach not only speeds up the division process but also reduces the potential for calculation errors.

Learning synthetic division is invaluable because it enhances your efficiency in dealing with polynomials significantly.

Polynomial Functions

Polynomial functions form the backbone of algebraic expressions and equations. These functions, like (4x^4 + 6x^3 + 4x^2 - 5x + 13) from our exercise, are composed of variables raised to whole number exponents and their coefficients. They can range from the simplest (a constant function) to the highly complex (involving many terms and high degrees).

In our example, the polynomial is of the fourth degree, which means the highest exponent is four. The general form can be written as (f(x) = ax^n + bx^{n-1} + ... + k), where (n) is non-negative and represents the highest power in the polynomial. Polynomial functions have certain properties:

• They have smooth, continuous curves when graphed.
• The degree of the polynomial gives us the maximum number of zeros or x-intercepts it may have.
• They're differentiable, which means we can calculate their rate of change or slope at any point.

Appreciating the structure of polynomial functions aids in understanding their graphical behavior and how they interact with other algebraic tools, such as division and the Remainder Theorem. Grasping these components enables students to solve polynomial equations and model real-world phenomena successfully.