Chapter 4

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(u\) is directly proportional to \(v\). If \(v=30\), then \(u=12\).

a) Sketch the graph of \(f .\) (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing or is decreasing. $$ f(x)=\frac{4}{x} $$

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ 3+2 i ; \quad \text { degree } 2 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(s\) varies directly as \(t\). If \(t=10\), then \(s=18\).

(a) Sketch the graph of \(f .\) (b) Find the domain \(D\) and range \(R\) of \(f\). (c) Find the intervals on which \(f\) is increasing or is decreasing. $$ f(x)=\frac{1}{x^{2}} $$

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ -4+3 i ; \quad \quad \text { degree } 2 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(r\) varies directly as \(s\) and inversely as \(t\). If \(s=-2\) and \(t=4\), then \(r=7\).

dentify any vertical asymptotes, horizontal asymptotes, and holes. $$ f(x)=\frac{-2(x+5)(x-6)}{(x-3)(x-6)} $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(w\) varies directly as \(z\) and inversely as the square root of \(u\). If \(z=2\) and \(u=9\), then \(w=6\).

Identify any vertical asymptotes, horizontal asymptotes, and holes. f(x)=\frac{2(x+4)(x+2)}{5(x+2)(x-1)}

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to the square of \(x\) and inversely proportional to the cube of \(z\). If \(x=5\) and \(z=3\), then \(y=25\).

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ -1,0,3+i ; \quad \text { degree } 4 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(q\) is inversely proportional to the sum of \(x\) and \(y\). If \(x=0.5\) and \(y=0.7\), then \(q=1.4\).

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(z\) is directly proportional to the product of the square of \(x\) and the cube of \(y\). If \(x=7\) and \(y=-2\), then \(z=16\).

Sketch the graph of \(f\). $$ f(x)=\frac{3}{x-4} $$

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ 4+3 i,-2+i ; \quad \text { degree } 4 $$

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). \(f(x)=9 x+4\) \(p(x)=2 x-5\)

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(r\) is directly proportional to the product of \(s\) and \(v\) and inversely proportional to the cube of \(p\). If \(s=2, v=3\), and \(p=5\), then \(r=40\).

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ 3+5 i,-1-i ; \quad \text { degree } 4 $$

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$ f(x)=7 x^{2}+3 x-10 ; \quad \quad p(x)=x^{2}-x+10 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to \(x\) and inversely proportional to the square of \(z\). If \(x=4\) and \(z=3\), then \(y=16\).

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ 0,-2 i, 1-i ; \quad \text { degree } 5 $$

$$ \text { Use the remainder theorem to find } f(c) \text {. } $$ $$ f(x)=3 x^{3}-x^{2}+5 x-4 ; \quad c=2 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to \(x\) and inversely proportional to the sum of \(r\) and \(s\). If \(x=3, r=5\), and \(s=7\), then \(y=2\).

Sketch the graph of \(f\). $$ f(x)=\frac{4 x}{2 x-5} $$

Exer. 1-10: A polynomial \(f(x)\) with real coefficients and leading coefficient 1 has the given zero(s) and degree. Express \(f(x)\) as a product of linear and quadratic polynomials with real coefficients that are irreducible over \(\mathbb{R}\). $$ 0,3 i, 4+i ; \quad \text { degree } 5 $$

$$ \text { Use the remainder theorem to find } f(c) \text {. } $$ $$ f(x)=2 x^{3}+4 x^{2}-3 x-1 ; \quad c=3 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to the square root of \(x\) and inversely proportional to the cube of \(z\). If \(x=9\) and \(z=2\), then \(y=5\).

Sketch the graph of \(f\). $$ f(x)=\frac{4 x-1}{2 x+3} $$

Exer. 11-14: Show that the equation has no rational root. $$ x^{3}+3 x^{2}-4 x+6=0 $$

$$ \text { Use the remainder theorem to find } f(c) \text {. } $$ $$ f(x)=x^{4}-6 x^{2}+4 x-8 ; \quad c=-3 $$

Express the statement as a formula that involves the given variables and a constant of proportionality \(k\), and then determine the value of \(k\) from the given conditions. \(y\) is directly proportional to the square of \(x\) and inversely proportional to the square root of \(z\). If \(x=5\) and \(z=16\), then \(y=10\).

Sketch the graph of \(f\). $$ f(x)=\frac{5 x+3}{3 x-7} $$

Exer. 11-14: Show that the equation has no rational root. $$ 3 x^{3}-4 x^{2}+7 x+5=0 $$

$$ \text { Use the remainder theorem to find } f(c) \text {. } $$ $$ f(x)=x^{4}+3 x^{2}-12 ; \quad c=-2 $$

The pressure \(P\) acting at a point in a liquid is directly proportional to the distance \(d\) from the surface of the liquid to the point. (a) Express \(P\) as a function of \(d\) by means of a formula that involves a constant of proportionality \(k\). (b) In a certain oil tank, the pressure at a depth of 2 feet is \(118 \mathrm{lb} / \mathrm{ft}^{2}\). Find the value of \(k\) in part (a). (c) Find the pressure at a depth of 5 feet for the oil tank in part (b). (d) Sketch a graph of the relationship between \(P\) and \(d\) for \(d \geq 0\).

Sketch the graph of \(f\). $$ f(x)=\frac{(4 x-1)(x-2)}{(2 x+3)(x-2)} $$

Exer. 11-14: Show that the equation has no rational root. $$ x^{5}-3 x^{3}+4 x^{2}+x-2=0 $$

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). f(x)=3 x^{3}-x^{2}+5 x-4 ; \quad c=2$

Hooke's law states that the force \(F\) required to stretch a spring \(x\) units beyond its natural length is directly proportional to \(x\). (a) Express \(F\) as a function of \(x\) by means of a formula that involves a constant of proportionality \(k\). (b) A weight of 4 pounds stretches a certain spring from its natural length of 10 inches to a length of \(10.3\) inches. Find the value of \(k\) in part (a). (c) What weight will stretch the spring in part (b) to a length of \(11.5\) inches? (d) Sketch a graph of the relationship between \(F\) and \(x\) for \(x \geq 0\).

Sketch the graph of \(f\). $$ f(x)=\frac{(5 x+3)(x+1)}{(3 x-7)(x+1)} $$

Exer. 11-14: Show that the equation has no rational root. $$ 2 x^{5}+3 x^{3}+7=0 $$

The electrical resistance \(R\) of a wire varies directly as its length \(l\) and inversely as the square of its diameter \(d\). (a) Express \(R\) in terms of \(l, d\), and a constant of variation \(k\). (b) A wire 100 feet long of diameter \(0.01\) inch has a resistance of 25 ohms. Find the value of \(k\) in part (a). (c) Sketch a graph of the relationship between \(R\) and \(d\) for \(l=100\) and \(d>0\). (d) Find the resistance of a wire made of the same material that has a diameter of \(0.015\) inch and is 50 feet long.

Sketch the graph of \(f\). $$ f(x)=\frac{x-2}{x^{2}-x-6} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ x^{3}-x^{2}-10 x-8=0 $$

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). f(x)=x^{4}-6 x^{2}+4 x-8 ; \quad c=-3$

The intensity of illumination \(I\) from a source of light varies inversely as the square of the distance \(d\) from the source. (a) Express \(I\) in terms of \(d\) and a constant of variation \(k\). (b) A searchlight has an intensity of \(1,000,000\) candlepower at a distance of 50 feet. Find the value of \(k\) in part (a). (c) Sketch a graph of the relationship between \(I\) and \(d\) for \(d>0 .\) (d) Approximate the intensity of the searchlight in part (b) at a distance of 1 mile.

Sketch the graph of \(f\). $$ f(x)=\frac{x+1}{x^{2}+2 x-3} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ x^{3}+x^{2}-14 x-24=0 $$

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). \(12 f(x)=x^{4}+3 x^{2}-12\) \(c=-2\)