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Chapter 3: Problem 120

Consider the following "monster" rational function. $$f(x)=\frac{x^{4}-3 x^{3}-21 x^{2}+43 x+60}{x^{4}-6 x^{3}+x^{2}+24 x-20}$$ Analyzing this function will synthesize many of the concepts of this and earlier chapters. Work Exercises \(119-128\) in order. Given that \(-4\) and \(-1\) are zeros of the numerator, factor the numerator completely.

Short Answer

Expert verified
The factored form of the numerator is (x + 4)(x + 1)(x - 3)(x - 5).

Step by step solution

01

Verify Given Zeros

Plug -4 and -1 into the numerator to verify that they are indeed zeros of the polynomial function. If substituting these values into the numerator (x^{4}-3x^{3}-21x^{2}+43x+60) results in zero, they are definitely zeros of the function.
02

Polynomial Division

Use synthetic division to divide the numerator, (x^4 - 3x^3 - 21x^2 + 43x + 60), by (x + 4) and (x + 1). Ensure you get zero as the remainder for both cases, which means (x + 4) and (x + 1) are factors.
03

First Synthetic Division (x + 4)

Perform synthetic division on (x^4 - 3x^3 - 21x^2 + 43x + 60) by (x + 4) to get a quotient polynomial of (x^3 - 7x^2 + 7x - 15).
04

Second Synthetic Division (x + 1)

Perform synthetic division again on the quotient polynomial (x^3 - 7x^2 + 7x - 15) by (x + 1) to get the new quotient polynomial (x^2 - 8x + 15).
05

Factor the Quadratic Polynomial

Factor the quadratic polynomial (x^2 - 8x + 15) further into (x - 3)(x - 5). Now the polynomial in the numerator is completely factored as (x + 4)(x + 1)(x - 3)(x - 5).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

rational functions
Rational functions are expressions that can be written as a ratio of two polynomials. For example, the given function is a rational function: \[ f(x) = \frac{x^4 - 3x^3 - 21x^2 + 43x + 60}{x^4 - 6x^3 + x^2 + 24x - 20} \] The numerator and the denominator are both polynomials. To analyze a rational function, you'll often need to find the zeros of the numerator and the denominator, as these will be key to understanding the behavior of the function. The zeros of the numerator are the values of x that make the numerator zero, and hence the function itself zero. The zeros of the denominator determine the vertical asymptotes, which are the values of x that make the function undefined.
synthetic division
Synthetic division is a simplified form of polynomial division, particularly useful for dividing by linear factors. Here's how you perform synthetic division using the zero \( x = -4 \) on the numerator polynomial \( x^4 - 3x^3 - 21x^2 + 43x + 60 \):
  • Write down the coefficients of the polynomial: \( 1, -3, -21, 43, 60 \)
  • Use \( -4 \) for the synthetic division, placing it outside the coefficients.
  • Bring down the first coefficient \( 1 \).
  • Multiply \( -4 \) by the value just written below the line \( 1 \), and add this product to the next coefficient.
  • Repeat the process for each coefficient.
  • The last number obtained should be zero if \( -4 \) is a zero of the polynomial.
This process results in a new polynomial. Repeat the division process for another zero, and you鈥檒l get a fully factored polynomial.
zeros of polynomials
The zeros of a polynomial are the values of \( x \) at which the polynomial evaluates to zero. To find the zeros of the given polynomial \( x^4 - 3x^3 - 21x^2 + 43x + 60 \), factorizing it is essential. Initially, we are given \( x = -4 \) and \( x = -1 \) as zeros.
  • Substitute these values into the polynomial to verify they produce zero.
  • Utilize these zeros to factorize the polynomial through synthetic division.
After factorizing the polynomial using the given zeros, you can determine additional zeros by factoring the resulting quotient polynomial further. In this case, factoring results in the zeros: \( x = -4, -1, 3, 5 \). These zeros are critical in graphing the polynomial and understanding its behavior.
factorization
Factorization is the process of breaking down a polynomial into simpler polynomials that, when multiplied together, give the original polynomial. To factorize the numerator of the given rational function, start by using the known zeros.
  • First, verify the zeros by plugging them into the polynomial.
  • Next, use synthetic division to divide the polynomial by factors of the form \( (x - \text{zero}) \).
  • Perform the division step-by-step until you reduce the polynomial to a factored form.
For the polynomial \( x^4 - 3x^3 - 21x^2 + 43x + 60 \), start with \( (x + 4) \) and \( (x + 1) \) to get intermediate polynomials. Continue factoring the resulting quadratic polynomial until fully factored into \( (x + 4)(x + 1)(x - 3)(x - 5) \). This complete factorization makes it easier to solve and analyze the polynomial.

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Most popular questions from this chapter

Let \(f\) be the function whose graph is obtained by translating the graph of \(y=\frac{1}{x}\) to the right 3 units and up 2 units. (a) Write an equation for \(f(x)\) as a quotient of two polynomials. (b) Determine the zero(s) of \(f\) (c) Identify the asymptotes of the graph of \(f(x)\)

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. A. The \(x\) -intercept is \(-3\) B. The \(y\) -intercept is 5 C. The horizontal asymptote is \(y=4\) D. The vertical asymptote is \(x=-1\) E. There is a "hole" in its graph at \(x=-4\) F. The graph has an oblique asymptote. G. The \(x\) -axis is its horizontal asymptote, and the \(y\) -axis is not its vertical asymptote. H. The \(x\) -axis is its horizontal asymptote, and the \(y\) -axis is its vertical asymptote. $$f(x)=\frac{1}{x+4}$$

Solve each problem. Height of an Object If an object is projected upward from ground level with an initial velocity of 32 ft per sec, then its height in feet after \(t\) seconds is given by $$ s(t)=-16 t^{2}+32 t $$ Find the number of seconds it will take to reach its maximum height. What is this maximum height?

The remainder theorem indicates that when a polynomial \(f(x)\) is divided by \(x-k\) the remainder is equal to \(f(k) .\) For $$f(x)=x^{3}-2 x^{2}-x+2$$ use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of \(f(x)\) $$f\left(\frac{3}{2}\right)$$

If \(c\) and \(d\) are complex numbers, prove each statement. (Hint: Let \(c=a+b i\) and \(d=m+n i\) and form all the conjugates, the sums, and the products.) $$\overline{c \cdot d}=\bar{c} \cdot \bar{d}$$
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