Question

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x⁵+2x⁴−15x³−10x²+12x+8

Accepted Answer

Start by finding the zeros of the polynomial function $$f(x) = 3x$$^{5}$$ + 2x$$^{4}$$ - 15x$$^{3}$$ - 10x$$^{2}$$ + 12x + 8. To do $$this, first look for possible rational zeros using the Rational Root Theorem. The possible rational zeros are of the form $$\pm \frac{p}{q}$$, where $$p$$ divides the constant term 8 and $$q$$ divides the leading coefficient 3. Test the possible rational zeros by substituting them into the polynomial or by using synthetic division to check if they yield a remainder of zero. Each zero found will correspond to a factor of the polynomial. Once a zero is found, use polynomial division (either long division or synthetic division) to divide the original polynomial by the corresponding factor $$(x - r)$$, where $$r is $$the zero. This will reduce the polynomial to a lower degree, making it easier to find the remaining zeros. Repeat the process of finding zeros and dividing the polynomial until you factor the polynomial completely into linear and/or irreducible quadratic factors. The zeros of the polynomial are the roots of these factors. After finding all zeros, analyze the multiplicity of each zero to understand the behavior of the graph at those points (whether the graph crosses or touches the x-axis). Then, use this information along with the end behavior of the polynomial (determined by the leading term $$3x^{5}) to $$sketch a complete graph of the function without using a graphing utility.

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x⁵+2x⁴−15x³−10x²+12x+8

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Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x5+2x4−15x3−10x2+12x+8

Risposta video verificata per un problema simile:

Concetti chiave

Finding Zeros of Polynomial Functions

End Behavior of Polynomial Functions

Sketching Polynomial Graphs Without Technology

Exercises 53–60 show incomplete graphs of given polynomial functions. a) Find all the zeros of each function. b) Without using a graphing utility, draw a complete graph of the function. f(x)=3x⁵+2x⁴−15x³−10x²+12x+8