Use Descartes' Rule of Signs to explain why 2x4+6x2+8=$$02x^4$$ + $$6x^2 + 8 = 0 $$has no real roots.

Ch. 3 - Polynomial and Rational Functions

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Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 39

Chapter 4, Problem 39

Use Descartes' Rule of Signs to explain why $2 x^{4} + 6 x^{2} + 8 = 0$ has no real roots.

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Write down the polynomial: $2x^4 + 6x^2 + 8 = 0$.

Identify the signs of the coefficients in the polynomial: all coefficients (2, 6, and 8) are positive.

Apply Descartes' Rule of Signs to the polynomial $f(x)$ by counting the number of sign changes in the sequence of coefficients. Since all coefficients are positive, there are 0 sign changes.

According to Descartes' Rule of Signs, the number of positive real roots is equal to the number of sign changes or less than that by an even number. Here, it means there are 0 positive real roots.

Next, apply Descartes' Rule of Signs to $f(-x)$ by substituting $-x$ into the polynomial and checking the signs of the coefficients. Since all powers of $x$ are even, $f(-x) = 2x^4 + 6x^2 + 8$ has the same coefficients and no sign changes, so there are 0 negative real roots as well.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Descartes' Rule of Signs

Descartes' Rule of Signs is a method to determine the possible number of positive and negative real roots of a polynomial by counting the sign changes in its coefficients. The number of positive real roots is either equal to the number of sign changes or less than it by an even number. To find negative roots, the rule is applied to the polynomial with x replaced by -x.

Polynomial Roots and Real Solutions

A root of a polynomial is a value of x that makes the polynomial equal to zero. Real roots are solutions that are real numbers, as opposed to complex or imaginary roots. Understanding the nature of roots helps in analyzing the behavior of polynomial equations and their graphs.

Sign Changes in Polynomial Coefficients

Sign changes occur when consecutive coefficients in a polynomial switch from positive to negative or vice versa. Counting these changes is essential in applying Descartes' Rule of Signs, as it directly relates to the maximum number of positive or negative real roots the polynomial can have.

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Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x³−4x²−7x+10

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Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x³−8x²+x+2; between 2 and 3

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For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. $f (x) = x^{3} + 3 x^{2} - 4$

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Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x⁴+10x³+5x²+x+1; f(−2/3)

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An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=3x²−12x−1

Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. $x^{3} + x^{2} + 4 x + 4 > 0$

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Use Descartes' Rule of Signs to explain why 2x4+6x2+8=02x^4 + 6x^2 + 8 = 0 has no real roots.

Verified video answer for a similar problem:

Key Concepts

Descartes' Rule of Signs

Polynomial Roots and Real Solutions

Sign Changes in Polynomial Coefficients

Use Descartes' Rule of Signs to explain why $2 x^{4} + 6 x^{2} + 8 = 0$ has no real roots.