Chapter 3

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=5 x^{2}+6 x^{3}$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(g\) varies directly as \(h\)

In Exercises \(1-8,\) find the domain of each rational function. $$f(x)=\frac{5 x}{x-4}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=x^{3}+x^{2}-4 x-4 $$

Use the Upper and Lower Bound Theorem to solve Exercises \(1-4\). Show that all the real roots of the equation \(x^{4}-5 x^{3}+11 x^{2}+33 x-18=0\) lie between \(-4\) and 7.

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\) $$\left(x^{2}+8 x+15\right) \div(x+5)$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=7 x^{2}+9 x^{4}$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(v\) varies directly as \(r\)

In Exercises \(1-8,\) find the domain of each rational function. $$f(x)=\frac{7 x}{x-8}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=x^{3}+3 x^{2}-6 x-8 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(x^{2}+3 x-10\right) \div(x-2)$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=7 x^{5}-\pi x^{3}+\frac{1}{5} x$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(a\) is directly proportional to the square of \(b\)

In Exercises \(1-8,\) find the domain of each rational function. $$g(x)=\frac{3 x^{2}}{(x-5)(x+4)}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=3 x^{4}-11 x^{3}-x^{2}+19 x+6 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(x^{3}+5 x^{2}+7 x+2\right) \div(x+2)$$

Use the Upper and Lower Bound Theorem to solve Exercises \(1-4\). Show that all the real roots of the equation \(2 x^{3}+5 x^{2}-8 x-7=0\) lie between \(-4\) and 2.

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(s\) is directly proportional to the cube of \(v\)

In Exercises \(1-8,\) find the domain of each rational function. $$g(x)=\frac{2 x^{2}}{(x-2)(x+6)}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=2 x^{4}+3 x^{3}-11 x^{2}-9 x+15 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(x^{3}-2 x^{2}-5 x+6\right) \div(x-3)$$

Use the Upper and Lower Bound Theorem to solve Exercises \(1-4\). Show that all the real roots of the equation \(2 x^{5}-13 x^{3}+2 x-5=0\) lie between \(-3\) and 3.

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(r\) varies inversely as \(t\)

In Exercises \(1-8,\) find the domain of each rational function. $$h(x)=\frac{x+7}{x^{2}-49}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=4 x^{4}-x^{3}+5 x^{2}-2 x-6 $$

Consider the equation \(x^{4}+3 x^{3}+2 x^{2}-5 x+12=0\) a. List all possible rational roots. b. Determine whether 1 is a root using synthetic division. What two conclusions can you draw? c. Based on part (b), what possible rational roots can you eliminate? d. Determine whether \(-3\) is a root using synthetic division. What two conclusions can you draw? e. Based on part (d), what possible rational roots can you eliminate?

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(6 x^{3}+7 x^{2}+12 x-5\right) \div(3 x-1)$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(w\) varies inversely as \(l\)

In Exercises \(1-8,\) find the domain of each rational function. $$h(x)=\frac{x+8}{x^{2}-64}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=3 x^{4}-11 x^{3}-3 x^{2}-6 x+8 $$

Consider the equation $$2 x^{5}+5 x^{4}-8 x^{3}-14 x^{2}+6 x+9=0$$ a. List all possible rational roots. b. Determine whether \(\frac{3}{2}\) is a root using synthetic division. What two conclusions can you draw? c. Based on part (b), what possible rational roots can you eliminate? d. Determine whether \(-3\) is a root using synthetic division. What two conclusions can you draw? e. Based on part (d), what possible rational roots can you eliminate?

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(6 x^{3}+17 x^{2}+27 x+20\right) \div(3 x+4)$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(a\) is inversely proportional to the cube of \(b\)

In Exercises \(7-14,\) show that each polynomial has a real zero between the given integers. Then use the Intermediate Value Theorem to find an approximation for this zero to the nearest tenth. If applicable, use a graphing utility's zero feature to verify your answer. \(f(x)=x^{3}-x-1 ;\) between 1 and 2

In Exercises \(1-8,\) find the domain of each rational function. $$f(x)=\frac{x+7}{x^{2}+49}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=x^{5}-x^{4}-7 x^{3}+7 x^{2}-12 x-12 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(12 x^{2}+x-4\right) \div(3 x-2)$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(y\) is inversely proportional to the square root of \(x .\)

In Exercises \(7-14,\) show that each polynomial has a real zero between the given integers. Then use the Intermediate Value Theorem to find an approximation for this zero to the nearest tenth. If applicable, use a graphing utility's zero feature to verify your answer. \(f(x)=x^{3}-4 x^{2}+2 ;\) between 0 and 1

In Exercises \(1-8,\) find the domain of each rational function. $$f(x)=\frac{x+8}{x^{2}+64}$$

Use the Rational Zero Theorem to list all possible rational zeros for each given function. $$ f(x)=4 x^{5}-8 x^{4}-x+2 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\left(4 x^{2}-8 x+6\right) \div(2 x-1)$$

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{x^{3}}$$

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(r\) varies directly as \(s\) and inversely as \(v\)

In Exercises \(7-14,\) show that each polynomial has a real zero between the given integers. Then use the Intermediate Value Theorem to find an approximation for this zero to the nearest tenth. If applicable, use a graphing utility's zero feature to verify your answer. \(f(x)=2 x^{4}-4 x^{2}+1 ;\) between \(-1\) and 0

a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the zero from part (b) to find all the zeros of the polynomial function. $$ f(x)=x^{3}+x^{2}-4 x-4 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{2 x^{3}+7 x^{2}+9 x-20}{x+3}$$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=2(x-3)^{2}+1\)

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=\frac{x^{2}+7}{3}$$