Chapter 3

Determine the constant of variation for each stated condition. \(D\) varies directly as \(E\) and inversely as \(F,\) and \(D=6\) when \(E=12\) and \(F=10\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ x^{4}-x^{3}-9 x^{2}+29 x-60=0 ; 1+2 i $$

a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the root from part (b) and solve the equation. $$ 2 x^{3}-5 x^{2}-6 x+4=0 $$

Divide using synthetic division. $$\left(5 x^{2}-12 x-8\right) \div(x+3)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=(x-3)^{2}+2\)

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=5 x^{3}+7 x^{2}-x+9$$

Determine the constant of variation for each stated condition. \(a\) varies jointly as \(b\) and \(c,\) and \(a=72\) when \(b=18\) and \(c=2\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ x^{4}-8 x^{3}+64 x-105=0 ; 2-i $$

a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the root from part (b) and solve the equation. $$ x^{4}-2 x^{3}-5 x^{2}+8 x+4=0 $$

In Exercises \(21-28,\) find the vertical asymptotes, if any, of the graph of each rational function. $$f(x)=\frac{x}{x+4}$$

Divide using synthetic division. $$\left(4 x^{3}-3 x^{2}+3 x-1\right) \div(x-1)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(y-1=(x-3)^{2}\)

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{3}-6 x^{2}+x+3$$

Determine the constant of variation for each stated condition. \(z\) varies jointly as \(w\) and \(y,\) and \(z=38\) when \(w=38\) and \(y=2\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ 4 x^{4}-28 x^{3}+129 x^{2}-130 x+125=0 ; 3-4 i $$

a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the root from part (b) and solve the equation. $$ x^{4}-2 x^{2}-16 x-15=0 $$

Find the vertical asymptotes, if any, of the graph of each rational function. $$f(x)=\frac{x}{x-3}$$

Divide using synthetic division. $$\left(5 x^{3}-6 x^{2}+3 x+11\right) \div(x-2)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(y-3=(x-1)^{2}\)

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=5 x^{4}+7 x^{2}-x+9$$

Use the four-step procedure for solving variation problems given on page 356 to solve. \(y\) varies directly as \(x . y=35\) when \(x=5 .\) Find \(y\) when \(x=12\).

In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that are irreducible over the real numbers. c. in completely factored form involving complex nonreal, or imaginary, numbers. $$ x^{4}-x^{2}-20 $$

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

Find the vertical asymptotes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x+4)}$$

Divide using synthetic division. $$\left(6 x^{5}-2 x^{3}+4 x^{2}-3 x+1\right) \div(x-2)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=2(x+2)^{2}-1\)

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{4}-6 x^{2}+x+3$$

In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that are irreducible over the real numbers. c. in completely factored form involving complex nonreal, or imaginary, numbers. $$ x^{4}+6 x^{2}-27 $$

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=x^{3}+7 x^{2}+x+7 $$

Find the vertical asymptotes, if any, of the graph of each rational function. $$g(x)=\frac{x+3}{x(x-3)}$$

Divide using synthetic division. $$\left(x^{5}+4 x^{4}-3 x^{2}+2 x+3\right) \div(x-3)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=\frac{5}{4}-\left(x-\frac{1}{2}\right)^{2}\)

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-5 x^{4}+7 x^{2}-x+9$$

In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that are irreducible over the real numbers. c. in completely factored form involving complex nonreal, or imaginary, numbers. $$ x^{4}+x^{2}-6 $$

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=5 x^{3}-3 x^{2}+3 x-1 $$

Find the vertical asymptotes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x+4)}$$

Divide using synthetic division. $$\left(x^{2}-5 x-5 x^{3}+x^{4}\right) \div(5+x)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=4-(x-1)^{2}\)

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=-11 x^{4}-6 x^{2}+x+3$$

In Exercises \(23-28\), factor each polynomial: a. as the product of factors that are irreducible over the rational numbers. b. as the product of factors that are irreducible over the real numbers. c. in completely factored form involving complex nonreal, or imaginary, numbers. $$ x^{4}-9 x^{2}-22 $$

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=-2 x^{3}+x^{2}-x+7 $$

Find the vertical asymptotes, if any, of the graph of each rational function. $$h(x)=\frac{x}{x(x-3)}$$

Divide using synthetic division. $$\left(x^{2}-6 x-6 x^{3}+x^{4}\right) \div(6+x)$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=1-(x-3)^{2}\)

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=2(x-5)(x+4)^{2}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. \(y\) is directly proportional to \(x\) and inversely proportional to the square of \(z . y=20\) when \(x=50\) and \(z=5 .\) Find \(y\) the when \(x=3\) and \(z=6\)

Use Descartes's Rule of Signs to determine the possible number of positive and negative real zeros for each given function. $$ f(x)=2 x^{4}-5 x^{3}-x^{2}-6 x+4 $$

Find the vertical asymptotes, if any, of the graph of each rational function. $$r(x)=\frac{x}{x^{2}+4}$$

Divide using synthetic division. $$\frac{x^{5}+x^{3}-2}{x-1}$$

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. \(f(x)=x^{2}-2 x-3\)