91Ó°ÊÓ

Use the intermediate value theorem for polynomials to show that each polynomial function has a real zero between the numbers given. $$f(x)=x^{4}-4 x^{3}-20 x^{2}+32 x+12 ;-1 \text { and } 0$$

Use synthetic division to decide whether the given number \(k\) is a zero of the given polynomial function. If it is not, give the value of \(f(k) .\) See Examples 2 and 3 . $$f(x)=x^{2}+3 x+4 ; k=2+i$$

Use synthetic division to decide whether the given number \(k\) is a zero of the given polynomial function. If it is not, give the value of \(f(k) .\) See Examples 2 and 3 . $$f(x)=x^{2}-3 x+5 ; k=1-2 i$$

Use the intermediate value theorem for polynomials to show that each polynomial function has a real zero between the numbers given. $$f(x)=x^{5}+2 x^{4}+x^{3}+3 ;-1.8 \text { and }-1.7$$

Use synthetic division to decide whether the given number \(k\) is a zero of the given polynomial function. If it is not, give the value of \(f(k) .\) See Examples 2 and 3 . $$f(x)=x^{3}+3 x^{2}-x+1 ; k=1+i$$

Show that the real zeros of each polynomial function satisfy the given conditions. \(f(x)=x^{4}-x^{3}+3 x^{2}-8 x+8 ;\) no real zero greater than 2

Show that the real zeros of each polynomial function satisfy the given conditions. \(f(x)=2 x^{5}-x^{4}+2 x^{3}-2 x^{2}+4 x-4 ;\) no real zero greater than 1

Use synthetic division to decide whether the given number \(k\) is a zero of the given polynomial function. If it is not, give the value of \(f(k) .\) See Examples 2 and 3 . $$f(x)=2 x^{3}-x^{2}+3 x-5 ; k=2-i$$

Show that the real zeros of each polynomial function satisfy the given conditions. \(f(x)=x^{4}+x^{3}-x^{2}+3 ;\) no real zero less than \(-2\)

The remainder theorem indicates that when a polynomial \(f(x)\) is divided by \(x-k\) the remainder is equal to \(f(k) .\) For $$f(x)=x^{3}-2 x^{2}-x+2$$ use the remainder theorem to find each of the following. Then determine the coordinates of the corresponding point on the graph of \(f(x)\) $$f(-2)$$