Question

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=3x4−2x3−8x+5f(x) = $$3x^4$$ - $$2x^3 - 8x + 5$$

Accepted Answer

Write down the polynomial function: $$f(x) = 3x^4$ - 2x^3$ - 8x + 5. To $$find the possible number of positive real zeros, count the number of sign changes in the coefficients of $$f(x). $$The coefficients are $$3$$, $$-2$$, $$0 ($$for $$x^2$$ term), $$-8$$, and $$5. $$Note that the zero coefficient does not affect sign changes. Identify the sign changes in $$f(x)$$: from $$3 to$$ $$-2 ($$positive to negative), from $$-2 to$$ $$0 (no $$sign change since zero is neutral), from $$0 to$$ $$-8 (no $$sign change), and from $$-8 to$$ $$5 ($$negative to positive). Count these sign changes to determine the possible number of positive real zeros. To find the possible number of negative real zeros, evaluate $$f(-x) by $$substituting $$-x$$ into the function: $$f(-x) = 3(-x)^4$ - 2(-x)^3$ - 8(-x) + 5. $$Simplify this expression to get a new polynomial. Count the number of sign changes in the coefficients of $$f(-x) to $$determine the possible number of negative real zeros. According to Descartes's Rule of Signs, the number of positive or negative real zeros is either equal to the number of sign changes or less than that by an even number.

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $f (x) = 3 x^{4} - 2 x^{3} - 8 x + 5$

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Key Concepts

Descartes's Rule of Signs

Polynomial Functions and Their Zeros

Evaluating f(-x) for Negative Zeros

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. f(x)=3x4−2x3−8x+5f(x) = 3x^4 - 2x^3 - 8x + 5