/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 59 Show that the given value(s) of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 59

Show that the given value(s) of \(c\) are zeros of \(P(x),\) and find all other zeros of \(P(x)\). $$P(x)=x^{3}-x^{2}-11 x+15, \quad c=3$$

Short Answer

Expert verified
The zeros of \( P(x) \) are 3, \(-1 + \sqrt{6}\), \(-1 - \sqrt{6}\).

Step by step solution

01

Verify given zero

To verify if a given value \( c \) is a zero of \( P(x) \), substitute \( c = 3 \) into the polynomial \( P(x) \) and evaluate it. If the result is zero, then \( c \) is indeed a zero.\[ P(3) = (3)^3 - (3)^2 - 11(3) + 15 \] Calculating further, we get:\[ P(3) = 27 - 9 - 33 + 15 = 0 \] Since \( P(3) = 0 \), \( c = 3 \) is a zero of \( P(x) \).
02

Perform polynomial division

Now that we know \( x - 3 \) is a factor of \( P(x) \), we perform polynomial division of \( P(x) \) by \( x - 3 \) to find the quotient polynomial.Performing synthetic division or long division, we divide:\[ P(x) \div (x - 3) \]The division yields:\[ x^2 + 2x - 5 \]
03

Solve the quadratic equation

Next, solve the quadratic equation \( x^2 + 2x - 5 = 0 \) to find the remaining zeros.We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \( a = 1, b = 2, c = -5 \). Substitute these values:\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \times 1 \times (-5)}}{2 \times 1} \]\[ x = \frac{-2 \pm \sqrt{4 + 20}}{2} \]\[ x = \frac{-2 \pm \sqrt{24}}{2} \]\[ x = \frac{-2 \pm 2\sqrt{6}}{2} \]\[ x = -1 \pm \sqrt{6} \]
04

Conclusion

Thus, the polynomial \( P(x) = x^3 - x^2 - 11x + 15 \) has the zeros \( c = 3 \), \( x = -1 + \sqrt{6} \), and \( x = -1 - \sqrt{6} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polynomial Division
Polynomial division is a process similar to long division that allows you to divide one polynomial by another, usually linear, polynomial. In our problem, we have a polynomial:
  • P(x), which is divided by x - 3 since we verified c = 3 is a zero of P(x).
To divide P(x), we could use either long division or synthetic division. With polynomial long division, you follow steps similar to numerical long division:
  • Divide the leading term of the dividend by the leading term of the divisor.
  • Multiply the entire divisor by that quotient.
  • Subtract the result from the original dividend.
  • Repeat the process with the remainder.
Once completed, if there is no remainder, it indicates that the divisor is a factor of the dividend.
In our example, after dividing by x - 3, the quotient obtained is a quadratic polynomial, x虏 + 2x - 5.
Quadratic Formula
The quadratic formula is a crucial tool for solving quadratic equations, which are of the form:\[ ax^2 + bx + c = 0 \]
The formula allows you to find the roots of any quadratic equation and is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, a, b, and c are coefficients from the polynomial.
In our exercise, after completing polynomial division, the remaining term was a quadratic equation:
  • \( x^2 + 2x - 5 = 0 \)
Plug the coefficients \(a = 1\), \(b = 2\), and \(c = -5\) into the quadratic formula:
  • Calculate the discriminant, \(b^2 - 4ac\), which indicates the nature of the roots.
  • Solve the formula to find the roots: \(-1 + \sqrt{6}\) and \(-1 - \sqrt{6}\).
This method provides an exact solution for calculating zeros of quadratic expressions regardless of complexity.
Synthetic Division
Synthetic division is a shortcut method specifically used for dividing a polynomial by a linear binomial of the form \(x - c\). It is quicker and simpler than polynomial long division, especially when working with polynomials of higher degrees.Begin by setting up the synthetic division process:
  • Write down the coefficients of the polynomial.
  • Use the zero (c, from \(x-c\)) outside the division symbol.
  • Bring down the leading coefficient to the bottom row.
  • Multiply it by c, add to the next coefficient, and repeat.
Each step generates the new coefficients of the quotient.
In our example, after confirming \(c = 3\) is a zero of the polynomial, you place 3 outside the synthetic division, use the coefficients from \(x^3 - x^2 - 11x + 15\), and perform the operation:
  • This gives the quotient polynomial \(x^2 + 2x - 5\),
  • showing that \(x - 3\) is a factor.
Synthetic division is most effective as it reduces the complexity of polynomial division while maintaining efficiency.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the maximum or minimum value of the function. $$f(x)=3-4 x-x^{2}$$

Find the slant asymptote and the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}-2 x-8}{x}$$

Maximum and Minimum Values A quadratic function \(f\) is given. (a) Express \(f\) in standard form. (b) Sketch a graph of \(f .\) (c) Find the maximum or minimum value of \(f .\) $$h(x)=1-x-x^{2}$$

At a certain vineyard it is found that each grape vine produces about 10 lb of grapes in a season when about 700 vines are planted per acre. For each additional vine that is planted, the production of each vine decreases by about 1 percent. So the number of pounds of grapes produced per acre is modeled by $$A(n)=(700+n)(10-0.01 n)$$ where \(n\) is the number of additional vines planted. Find the number of vines that should be planted to maximize grape production.

Find the maximum or minimum value of the function. $$g(x)=2 x(x-4)+7$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.