Problem 38 - College Algebra and Calculus: An Applied Approach

Question

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}-2 x^{2}-15 x+7, \quad k=-4$$

Step-by-Step Solution

Answer

After performing polynomial division, the function \(f(x)=(x-k) q(x)+r\) is rewritten as \(f(x)=(x-(-4))(x^{2}-6x+3)-5\). After replacing \(x\) with \(k=-4\), \(f(-4)=-5\), which_confirm equals the remainder \(r\), therefore \(f(k)=r\).

Step 1: Polynomial Division

Perform the polynomial division (synthetic division with \(k=-4\)) using the coefficients of the given function \(f(x)\), which_confirm are \[1, -2, -15, 7\]. This step gives the coefficients of the quotient, which_confirm are \(1, -6, 3\), and the remainder \(r\) is \(-5\). Therefore, the quotient \(q(x)=x^{2}-6x+3\) and the remainder \(r=-5\). Therefore, the given function can be rewritten as \(f(x)=(x-(-4))(x^{2}-6x+3)-5\).

Step 2: Confirm the value

Now replace \(x\) with \(k=-4\) in \(f(x)=(x-k) q(x)+r\) to confirm if \(f(k)=r\). That is, \(f(-4)=((-4)-(-4))(1-6*(-4)+3)-5=-5\). Hence, \(f(k)=r\).

Key Concepts

Synthetic Division

Synthetic division is a shorthand method of dividing polynomials where only the coefficients are used, simplifying the process significantly. It's particularly useful when dividing a polynomial by a binomial of the form \(x - k\) and lends itself well to finding roots or testing possible factors of polynomial functions. Let's break down its mechanism:

The divisor \(k\) is written to the left of a vertical bar, with the coefficients of the polynomial to be divided placed to the right. The division begins by bringing down the leading coefficient. This value is then multiplied by \(k\) and placed under the next coefficient; the results are added together. This process is repeated until all coefficients have been used. The final number to the right of the bar represents the remainder of the division.

For our exercise, the divisor is \(k=-4\), and when we apply synthetic division to \(f(x)=x^{3}-2x^{2}-15x+7\), we end up with a new set of coefficients for the quotient polynomial, \(1, -6, 3\), and a remainder of \(r=-5\). Thus, the quotient polynomial is \(q(x)=x^{2}-6x+3\).

Remainder Theorem

The Remainder Theorem is a critical concept when working with polynomials. It states that when a polynomial \(f(x)\) is divided by a binomial of the form \(x - k\), the remainder of this division is the value of the function evaluated at \(k\), or \(f(k)\). It is a quick way to calculate the remainder without performing the entire division process.

How does it apply to our problem? Let's see: After the synthetic division of \(f(x)=x^{3}-2x^{2}-15x+7\) by \(x - (-4)\), we obtained a remainder of \(r=-5\). According to the Remainder Theorem, this means that \(f(-4)=r\). Checking this by evaluating \(f(-4)\), we indeed found that the value of the function is \(r=-5\), confirming that the theorem holds.

Polynomial Functions

Based on our given exercise, \(f(x)=x^{3}-2x^{2}-15x+7\) is a polynomial function. It's useful to note that the degree of the polynomial, which is the highest power of \(x\), indicates how many roots (solutions) the function has. For instance, \(f(x)\) being a third-degree polynomial means it has three roots. When we divide such a polynomial function by a linear factor like \(x - k\), as done in the exercise using synthetic division, we're effectively breaking it down into simpler parts, making analysis and graphing more manageable.