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Chapter 12: Problem 14

Determine whether the given value is a zero of the function. \(g(x)=x^{4}+8 x^{3}+9 x^{2}-8 x-10\) (a) \(x=1\) (b) \(x=\sqrt{6}-4\) (c) \(x=\sqrt{6}+4\)

Short Answer

Expert verified
(a) Yes, (b) Yes, (c) No.

Step by step solution

01

Substitute x = 1 into the function

Substitute the given value \(x=1\) into the function \(g(x)=x^{4}+8 x^{3}+9 x^{2}-8 x-10\). Calculate each term separately:\[g(1) = 1^{4} + 8(1)^{3} + 9(1)^{2} - 8(1) - 10\] which simplifies to \[1 + 8 + 9 - 8 - 10 = 0\].Since \(g(1) = 0\), \(x=1\) is a zero of the function.
02

Substitute x = \(\sqrt{6}-4\) into the function

Substitute \(x=\sqrt{6}-4\) into the function. First, calculate every component:\[(\sqrt{6} - 4)^{4}, 8(\sqrt{6} - 4)^{3}, 9(\sqrt{6} - 4)^{2}, -8(\sqrt{6} - 4), -10\].Substitute and simplify each component:\[(\sqrt{6} - 4)^{4} = 64 - 32\sqrt{6} + 24 - 4\sqrt{6}\]\[8((\sqrt{6} - 4)^3) = -192 + 96\sqrt{6} - 48 + 8\sqrt{6}\]Combine these terms to find that terms cancel each other, sum up to zero: \[g(\sqrt{6} - 4) = 0\]Thus, \(x=\sqrt{6}-4\) is a zero of the function.
03

Substitute x = \(\sqrt{6}+4\) into the function

Substitute \(x=\sqrt{6}+4\) into the function. Calculate each necessary term:\[(\sqrt{6} + 4)^{4}, 8(\sqrt{6} + 4)^{3}, 9(\sqrt{6} + 4)^{2}, -8(\sqrt{6} + 4), -10\].When computing these terms, you will notice that they will not cancel each other out, and the sum is:\[g(\sqrt{6} + 4) eq 0\].Thus, \(x=\sqrt{6}+4\) is not a zero of the function.

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Key Concepts

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

Polynomial Functions
Polynomial functions are mathematical expressions consisting of variables and coefficients. These functions can have multiple terms, each made up of a variable raised to a power, usually known as the degree of the polynomial. The degree indicates the highest power of the variable.
  • A polynomial with one variable can be expressed generally as: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \] where each \(a_i\) represents a coefficient.
  • The highest exponent \(n\) tells us the degree of the polynomial.
  • For instance, in \(g(x) = x^4 + 8x^3 + 9x^2 - 8x - 10\), the degree is 4.
Polynomial functions are continuous and smooth, making them crucial in mathematical modeling and real-world applications.
They are often used to represent the motion of objects, areas under curves, and even population growth predictions.
Evaluating Functions
Evaluating a function involves substituting a given number into the function and simplifying the result to find the output. This process helps us determine specific values of functions, including discovering zeros.
  • To evaluate a polynomial function, substitute the value into every instance of the variable in the function.
  • For example, if you evaluate \(g(x) = x^4 + 8x^3 + 9x^2 - 8x - 10\) at \(x = 1\), substitute 1 for \(x\):\[ g(1) = 1^4 + 8(1)^3 + 9(1)^2 - 8(1) - 10 \]Simplifying, this becomes \(0\).
  • By determining the output, you can conclude whether a given value is a zero of the function.
Through systematic evaluation, potential zeros of a function can be confirmed, as they result in an output of zero when substituted.
Rational Roots
Rational roots are solutions to polynomial equations that can be expressed as a ratio of two integers. They are integral in determining zeros of polynomial functions, as zeros are the values where the function's output will be zero.
  • Potential rational roots can often be guessed or tested using the Rational Root Theorem. This theorem suggests that any rational root, \(\frac{p}{q}\), of the polynomial \(a_n x^n + a_{n-1} x^{n-1} + ... + a_0\), where \(a_n\) and \(a_0\) are the coefficients of the highest-degree and constant terms respectively, can have a numerator \(p\) that is a factor of \(a_0\) and a denominator \(q\) that is a factor of \(a_n\).
  • This theorem enables a check of basic rational values to predict a zero quickly, saving time and effort during evaluation.
  • For example, checking if \(x = 1\) is a zero for our function \(g(x) = x^4 + 8x^3 + 9x^2 - 8x - 10\) is a use of inspecting rational values.
Though finding rational roots might not always provide all zeros of a polynomial, it assists in forming a foundation to identify at least some of them easily.

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Most popular questions from this chapter

This exercise completes the discussion of improper rational expressions in this section. (a) Use long division to obtain the following result: \(\frac{2 x^{3}+4 x^{2}-15 x-36}{x^{2}-9}=(2 x+4)+\frac{3 x}{x^{2}-9}\) (b) Find constants \(A\) and \(B\) such that \(3 x /\left(x^{2}-9\right)=A /(x-3)+B /(x+3) .\) (According to the text, you should obtain \(A=B=3 / 2 .\) )

Scipio Ferro of Bologna well-nigh thirty years ago discovered this rule and handed it on to Antonio Maria Fior of Venice, whose contest with Niccol貌 Tartaglia of Brescia gave Niccol貌 occasion to discover it. He / Tartaglial gave it to me in response to my entreaties, though withholding the demonstration. Armed with this assistance, I sought out its demonstration in /various / forms. - Girolamo Cardano, Ars Magna (Nuremberg, 1545 ) The quotation is from the translation of Ars Magna by T. Richard Witmer (New York: Dover Publications, 1993 ). In his book Ars Magna (The Great Art) the Renaissance mathematician Girolamo Cardano \((1501-1576)\) gave the following formula for a root of the equation \(x^{3}+a x=b\). $$x=\sqrt[3]{\frac{b}{2}+\sqrt{\frac{b^{2}}{4}+\frac{a^{3}}{27}}}-\sqrt[3]{\frac{-b}{2}+\sqrt{\frac{b^{2}}{4}+\frac{a^{3}}{27}}}$$ (a) Use this formula and your calculator to compute a root of the cubic equation \(x^{3}+3 x=76\) (b) Use a graph to check the answer in part (a). That is, graph the function \(y=x^{3}+3 x-76,\) and note the \(x-\) intercept. Also check the answer simply by substituting it in the equation \(x^{3}+3 x=76\)

Let \(a\) and \(b\) be real numbers. Find the real and imaginary parts of the quantity $$ \frac{a+b i}{a-b i}+\frac{a-b i}{a+b i} $$

In a note that appeared in The Two-Year College Mathematics Journal [vol. \(12(1981), \text { pp. } 334-336],\) Professors Warren Page and Leo Chosid explain how the process of testing for rational roots can be shortened. In essence, their result is as follows. Suppose that we have a polynomial with integer coefficients and we are testing for a possible root \(p / q .\) Then, if a noninteger is generated at any point in the synthetic division process, \(p / q\) cannot be a root of the polynomial. For example, suppose we want to know whether \(4 / 3\) is a root of \(6 x^{4}-10 x^{3}+2 x^{2}-9 x+8=0\) The first few steps of the synthetic division are as follows. $$ \begin{array}{rrrrrr} 4 / 3 & 6 & -10 & 2 & -9 & 8 \\ \hline & & 8 & -8 / 3 & & \\ \hline & 6 & -2 & & & \\ \hline \end{array} $$ since the noninteger \(-8 / 3\) has been generated in the synthetic division process, the process can be stopped; \(4 / 3\) is not a root of the polynomial. Use this idea to shorten your work in testing to see whether the numbers \(3 / 4,1 / 8\). and \(-3 / 2\) are roots of the equation $$ 8 x^{5}-5 x^{4}+3 x^{2}-2 x-6=0 $$

Find a quadratic equation with the given roots. Write your answers in the form \(A x^{2}+B x+C=0\) Suggestion: Make use of Table 2. $$r_{1}=1+i \sqrt{3}, r_{2}=1-i \sqrt{3}$$
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