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Chapter 4: Problem 43

Given that the polynomial function has the given zero. find the other zeros. $$f(x)=x^{3}+5 x^{2}-2 x-10 ;-5$$

Short Answer

Expert verified
The other zeros are \ +\root2, -\root2 \.

Step by step solution

01

Verify the given zero

Substitute the given zero \(x = -5\) into the polynomial \(f(x)=x^{3}+5 x^{2}-2 x-10\) to verify it is indeed a zero: \(f(-5) = (-5)^3 + 5(-5)^2 - 2(-5) - 10\). Calculate: \ f(-5) = -125 + 125 + 10 - 10 = 0\. Since \(f(-5) = 0\), \(-5\) is confirmed as a zero.
02

Perform Synthetic Division

Perform synthetic division using the zero \(x = -5\) to divide the polynomial \(f(x)\). Coefficients of \(f(x)\) are \[1, 5, -2, -10\]. -5 | \quad 1 \quad 5 \quad -2 \quad -10 .. | \quad \downarrow \quad -5 \quad \quad 0 \quad 10 -------------------------------- \quad 1 \quad 0 \quad -2 \quad 0 Quotient is \(x^2 - 2\).
03

Solve the Quotient Polynomial

Solve the quadratic equation obtained from the quotient, \(x^2 - 2 = 0\) by setting each factor equal to zero. \[x^2 - 2 = 0\] Add 2 to both sides: \[x^2 = 2\] Take the square root of both sides: \[x = \pm \sqrt{2}\]. The other zeros are \(+\root{2}, -\root{2}\).

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Key Concepts

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

polynomial functions
A polynomial function is an expression made up of variables (or indeterminates) and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. For example, in the exercise, the polynomial function given is: \(f(x) = x^{3} + 5x^{2} - 2x - 10\) This polynomial is cubic because the highest exponent is 3. Understanding polynomial functions is essential as they are foundational in algebra and calculus.
synthetic division
Synthetic division is a shortcut method to divide a polynomial by a binomial of the form \((x - c)\). It simplifies the process and is less labor-intensive than traditional long division. Here鈥檚 how we used it in the exercise: We started with the polynomial \(f(x)\) and the given zero, \(x = -5\). By placing '-5' and the coefficients of the polynomial, we perform synthetic division:
  • Write down the coefficients: [1, 5, -2, -10]
  • Use the zero (-5) and perform the synthetic division steps
  • Find the quotient: \(x^2 - 2\)
This results in a simplified polynomial that is easier to solve.
quadratic equations
A quadratic equation is a second-degree polynomial and has the standard form: \(ax^2 + bx + c = 0\). In the exercise, after performing synthetic division, the quotient was a quadratic equation: \(x^2 - 2\). To find the other zeros, we solved this quadratic equation: \(x^2 - 2 = 0\)
  • Add 2 to both sides: \(x^2 = 2\)
  • Take the square root of both sides: \(x = \pm \sqrt{2}\)
Thus, the remaining zeros of the polynomial are \(+\sqrt{2}\) and \(-\sqrt{2} \).
zero theorem
The Zero Theorem states that for a polynomial \(f(x)\), if \(c\) is a zero, then \(f(c) = 0\). We use this theorem to find the zeros of polynomial functions and verify them. In the exercise: The given zero was \(x = -5\). We verified this by substituting -5 into the function: \(f(-5) = (-5)^3 + 5(-5)^2 - 2(-5) - 10 = -125 + 125 + 10 - 10 = 0\) Since \(f(-5) = 0\), -5 is indeed a zero of the polynomial. This theorem helps us validate whether a given value is a zero of a polynomial function, ensuring the division and further calculations are correct.

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Most popular questions from this chapter

Write a polynomial inequality for which the solution set is \([-4,3] \cup[7, \infty)\)

Fill in the blank with the correct term. Some of the given choices will not be used. Others will be used more than once. $$\begin{array}{ll}x \text { -intercept } & \text { midpoint formula } \\ y \text { -intercept } & \text { horizontal lines } \\ \text { odd function } & \text { vertical lines } \\ \text { even function } & \text { point-slope equation } \\ \text { domain } & \text { slope-intercept equation } \\ \text { range } & \text { difference quotient } \\ \text { slope } & f(x)=f(-x) \\\ \text { distance formula } & f(-x)=-f(x)\end{array}$$ A(n) ________ is a point \((0, b)\)

Fill in the blank with the correct term. Some of the given choices will not be used. Others will be used more than once. $$\begin{array}{ll}x \text { -intercept } & \text { midpoint formula } \\ y \text { -intercept } & \text { horizontal lines } \\ \text { odd function } & \text { vertical lines } \\ \text { even function } & \text { point-slope equation } \\ \text { domain } & \text { slope-intercept equation } \\ \text { range } & \text { difference quotient } \\ \text { slope } & f(x)=f(-x) \\\ \text { distance formula } & f(-x)=-f(x)\end{array}$$ The _____ of the line with slope \(m\) passing through \(\left(x_{1}, y_{1}\right)\) is \(y-y_{1}=m\left(x-x_{1}\right)\).

Use synthetic division and the remainder theorem to find the zeros. $$f(x)=-x^{3}+3 x^{2}+6 x-8$$

Make a hand-drawn graph. Be sure to label all the asymptotes. List the domain and the \(x\) -intercepts and the \(y\) -intercepts. Check your work using \(a\) graphing calculator. $$f(x)=\frac{5 x^{4}}{x^{4}+1}$$
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