Consider the polynomial function g(x)=−x5−9x3−8x. Factor g(x) completely and determine all of its complex zeros. Which of the zeros of g are not x-intercepts of the graph of y=g(x)? Why do these zeros not correspond to a point on the x-axis? In general, what can be said about the similarities and differences between the zeros and the x-intercepts of the graphs of polynomial functions?

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Consider the polynomial function g(x)=−x5−9x3−8x. Factor g(x) completely and determine all of its complex zeros. Which of the zeros of g are not x-intercepts of the graph of y=g(x)? Why do these zeros not correspond to a point on the x-axis? In general, what can be said about the similarities and differences between the zeros and the x-intercepts of the graphs of polynomial functions?

tacalaohn · Accepted Answer

Step 1  Given g(x)=−x5−9x3−8x  To find the zeros we put g(x)=0  −x5−9x3−8x=0  −x(x4+9x2+8)=0  −x(x4+8x2+x2+8)=0  −x(x2(x2+8)+1(x2+8))=0  −x(x2+1)(x2+8)=0  The zeros are x=0,x2+1=0andx2+8=0  x2+1=0⇒x2=−1⇒x=±i  and x2+8=0⇒x2=−8⇒x+±22i  Hence the zeros are x=0,x=±i, and x=±22i  Step 2  x=±i and x=±22i are the two roots which are not the x-intercepts of the graph y=g(x).  This is because the above mentioned 4 roots are imaginary or complex roots in nature which cannot be defined in the real plane hence these cannot be included in the x-intercepts.  Step 3  In general whenever any polynomial function have real roots or zeros then the zeros are also the x-intercept but when the roots of a polynomial function are imaginary or complex then those complex points cannot be represented in a real number line or graph or plane due to which these cannot be the x-intercept.

Consider the polynomial function g(x) = -x^5 - 9x^3 - 8x. Factor g(x) completely and determine

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