Chapter 3

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=2 x^{3}-x^{2}+6 x-3 ; x=\frac{1}{2}$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}-2$$

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{-2 x^{2}}{x-1}$$

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$x^{2}+x+1 ; x-1$$

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$f(s)=4 s^{5}-5 s^{3}+6 s-1$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$h(x)=2 x^{4}-x^{2}+2$$

Solve the polynomial inequality. $$(x+2)\left(x^{2}-4 x+5\right) \geq 0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=3 x^{3}+x^{2}+24 x+8 ; x=-\frac{1}{3}$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}-\sqrt{3}$$

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{2 x+7}{2 x^{2}+5 x-3}$$

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$3 x^{3}+2 x-8 ; x-4$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$f(x)=-3 x^{3}+1$$

Solve the polynomial inequality. $$(x+3)\left(x^{2}-3 x+2\right) \geq 0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=2 x^{5}+x^{4}-2 x-1 ; x=-\frac{1}{2}$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}-5$$

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{3 x+5}{x^{2}-x-2}$$

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$4 x^{3}-x+4 ; x-2$$

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$g(t)=\frac{1}{t}$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=x^{3}-2$$

Solve the polynomial inequality. $$x^{4}-x^{2}>3$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}+9$$

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{x+1}{x^{2}+1}$$

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$x^{6}-3 x^{5}+x^{4}-2 x^{2}-5 x+6 ; x^{2}+2$$

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$f(x)=5$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$f(x)=-x^{3}+2 x$$

Solve the polynomial inequality. $$x^{4}-3 x^{2}<10$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}+4$$

Find the domain and the vertical and horizontal asymptotes (if any). $$h(x)=\frac{x+2}{4+x^{2}}$$

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$-x^{6}+4 x^{5}-x^{3}+x^{2}+x-8 ; x^{2}+4$$

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$g(x)=-2$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=x^{3}-3 x$$

Solve the polynomial inequality. $$x^{3}-4 x \leq-x^{2}+4$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. \(p(x)=x^{4}-9\) (Hint: Factor first as a difference of squares.)

Use synthetic division to find the function values. \(f(x)=x^{3}-7 x+5 ;\) find \(f(3)\) and \(f(5)\)

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$f(x)=-(x+1)^{3}$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$h(x)=-2 x^{4}+3 x^{2}-1$$

Solve the polynomial inequality. $$x^{3}-7 x \leq-6$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. \(p(x)=x^{4}-16\) (Hint: Factor first as a difference of squares.)

Use synthetic division to find the function values. \(f(x)=-2 x^{3}+4 x^{2}-7 ;\) find \(f(4)\) and \(f(-3)\)

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$g(x)=(x-1)^{2}$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=3 x^{4}-2 x^{2}+1$$

Find all the real zeros of the polynomial. $$P(x)=x^{3}+2 x^{2}-5 x-6$$

Solve the polynomial inequality. $$x^{3} \leq 4 x$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear and irreducible quadratic factors. $$x^{3}-2 x^{2}+x-2 ; \text { zero: } x=2$$

Use synthetic division to find the function values. \(f(x)=-2 x^{4}-10 x^{3}-3 x+10 ;\) find \(f(-1)\) and \(f(2)\)

Determine the end behavior of the function. $$f(t)=7 t$$

Find all the real zeros of the polynomial. $$P(x)=2 x^{3}+3 x^{2}-8 x+3$$

Solve the polynomial inequality. $$x^{3} \geq x$$

One zero of each polynomial is given. Use it to express the polynomial as a product of linear and irreducible quadratic factors. $$x^{3}-x^{2}+4 x-4 ; \text { zero: } x=1$$

Use synthetic division to find the function values. \(f(x)=-x^{4}+3 x^{3}-2 x-4 ;\) find \(f(-2)\) and \(f(3).\)