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Chapter 2: Problem 70

(a) verify the given factors of \(f(x),\) (b) find the remaining factor(s) of \(f(x),(\mathrm{c})\) use your results to write the complete factorization of \(f(x),(d)\) list all real zeros of \(f,\) and (e) confirm your results by using a graphing utility to graph the function. Function \(\quad\) Factors \(f(x)=8 x^{4}-14 x^{3}-71 x^{2}-10 x+24 \qquad (x+2),(x-4)\)

Short Answer

Expert verified
The complete factorization of \(f(x)\) is \(f(x) = (x+2)(x-4)(8x^2 -6x -3)\). The real zeros of \(f(x)\) are \(x = -2, -0.75, 0.5, 4\).

Step by step solution

01

Verifying the given factors

First, we can check if \(x+2\) and \(x-4\) are indeed factors of \(f(x)\) by plugging them into the function. If they output 0, then they are indeed factors by the factor theorem. Doing so, we find \(f(-2) = 8(-2)^4 - 14(-2)^3 - 71(-2)^2 - 10(-2) + 24 = 0\) and \(f(4) = 8(4)^4 - 14(4)^3 - 71(4)^2 - 10(4) + 24 = 0\). Therefore, \(x+2\) and \(x-4\) are factors of \(f(x)\).
02

Finding the remaining factors

To find the remaining factors, we perform polynomial division or synthetic division of \(f(x)\) with \(x+2\) and \(x-4\). Doing so, we find that the remaining factor of \(f(x)\) is \(8x^2 -6x -3\).
03

Writing the complete factorization

Now that we have the factors, the complete factorization of the polynomial can be written as \(f(x) = (x+2)(x-4)(8x^2 -6x -3)\).
04

Listing all real zeros

The real zeros of the function can be found by setting all the factors equal to 0 and solving for \(x\). Doing so, we get \(x = -2, 4\) from the factors \(x+2, x-4\). And from the quadratic equation \(8x^2 -6x -3 = 0\), we find \(x = 0.5\) and \(x = -0.75\). Therefore, the real zeros of the function are \(x = -2, -0.75, 0.5, 4\).
05

Confirming results using a graphing utility

By graphing our function on a graphing tool, we can see that the function \(f(x) = 8x^4 -14x^3 -71x^2 -10x + 24\) touches the x-axis at \(x = -2, -0.75, 0.5, 4\), which matches our previously found zeros. Therefore, the given factors and the found zeros and factors are the correct ones.

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Key Concepts

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

The Factor Theorem
The Factor Theorem is a powerful tool in algebra that helps us identify whether a given expression is a factor of a polynomial. It states that a polynomial \( f(x) \) has a factor \((x - c)\) if and only if \( f(c) = 0 \). In simpler terms, if plugging a number, \( c \), into the polynomial equation results in zero, then \( (x - c) \) is a factor.

For example, suppose you want to verify if \( (x + 2) \) is a factor of the polynomial \( f(x) = 8x^4 - 14x^3 - 71x^2 - 10x + 24 \). You would substitute \( -2 \) for \( x \) in the polynomial. If \( f(-2) = 0 \), as it turns out it does, then \( (x + 2) \) is indeed a factor according to the Factor Theorem.
  • Plug potential zeros into the polynomial.
  • If \( f(c) = 0 \), \((x - c)\) is a factor.
  • This method helps find rational zeros quickly.
Polynomial Division
Polynomial Division is a technique used to divide two polynomials, similar to long division with numbers. It's a systematic method to reduce a polynomial, revealing its simpler factors. When dividing a polynomial \( f(x) \) by \( (x - c) \), we aim to find another polynomial \( q(x) \) as a quotient and possibly a remainder
which should be zero if \( (x - c) \) is truly a factor.

To divide \( f(x) = 8x^4 - 14x^3 - 71x^2 - 10x + 24 \) by \( (x + 2) \) or any similar factor, you follow these steps:
  • Divide the leading term of \( f(x) \) by the leading term of \( (x + 2) \).
  • Multiply the entire divisor by the result from the previous step.
  • Subtract the resulting polynomial from \( f(x) \).
  • Repeat the process with the new polynomial remainder until you can't divide anymore.
This method can seem tricky at first, but practice makes it easier and quicker. It鈥檚 very useful when combined with the Factor Theorem to simplify complex polynomials.
Synthetic Division
Synthetic Division is an efficient shortcut for dividing a polynomial by a linear factor, specifically in the form \( (x - c) \). It simplifies the process, making it faster and less cumbersome than long division.

Here's how you perform synthetic division between a polynomial like \( f(x) = 8x^4 - 14x^3 - 71x^2 - 10x + 24 \) and \( (x + 2) \):
  • Write down the coefficients of the polynomial.
  • Bring down the first coefficient to the bottom row.
  • Multiply this number by \( c \), which is the zero of \( (x - c) \) or, in this case, \(-2\).
  • Add the result to the next coefficient and bring the result down.
  • Repeat until all coefficients have been used.
  • If done correctly, the final row represents the coefficients of the quotient, and the last number (should be zero if \( (x - c) \) is a factor) is the remainder.
Synthetic division simplifies the polynomial division process considerably, especially with a pencil and paper, maintaining accuracy and speed.
Real Zeros
Real Zeros of a polynomial are the points where the graph of the polynomial crosses or touches the x-axis. In more technical terms, they are the real number solutions to the equation \( f(x) = 0 \). Finding the real zeros of a polynomial provides vital information about its behavior and graph.

For the polynomial \( f(x) = 8x^4 - 14x^3 - 71x^2 - 10x + 24 \), once we determine the factors, we identify the real zeros by setting each factor equal to zero:
  • For \( (x + 2) = 0 \), \( x = -2 \) is a zero.
  • For \( (x - 4) = 0 \), \( x = 4 \) is a zero.
  • The quadratic factor \( 8x^2 - 6x - 3 = 0 \) involves solving a quadratic equation, yielding \( x = 0.5 \) and \( x = -0.75 \).
These zeros provide a complete picture of where the polynomial's graph intersects the x-axis, allowing us to understand the polynomial's roots and overall behavior.

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Most popular questions from this chapter

Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=-\frac{1}{3} x^{2}+3 x-6$$

The profit \(P\) (in hundreds of dollars) that a company makes depends on the amount \(x\) (in hundreds of dollars) the company spends on advertising according to the model \(P=230+20 x-0.5 x^{2} . \quad\) What expenditure for advertising will yield a maximum profit?

Write the equation for a quadratic function \(f\) (with integer coefficients) that has the given zeros. Assume that \(b\) is a positive integer. (a) \(\pm \sqrt{b} i\) (b) \(a \pm b i\)

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Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of symmetry, and \(x\) -intercept(s). $$f(x)=x^{2}-x+\frac{5}{4}$$
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