Chapter 3

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) and then determine the value of \(k\) from the given conditions. \(u\) is directly proportional to \(v .\) If \(v=30,\) then \(u=12\)

A 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 \(\mathbb{R}\). $$3+2 i; \quad \text { degree } 2$$

(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. $$f(x)=\frac{4}{x}$$

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-1,2,3 ; \quad f(-2)=80$$

Sketch the graph of \(f\) for the indicated value of \(c\) or \(a\) $$f(x)=2 x^{3}+c$$ (a) \(c=3\) (b) \(c=-3\)

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=2 x^{4}-x^{3}-3 x^{2}+7 x-12 ; \quad p(x)=x^{2}-3$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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. $$f(x)=\frac{1}{x^{2}}$$

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-5,2,4 ; \quad f(3)=-24$$

Sketch the graph of \(f\) for the indicated value of \(c\) or \(a\) $$f(x)=-2 x^{3}+c$$ (a) \(c=-2\) (b) \(c=2\)

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=3 x^{4}+2 x^{3}-x^{2}-x-6 ; \quad p(x)=x^{2}+1$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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 \(r=7\)

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 \(\mathbb{R}\). $$2,-2-5 i; \quad \text { degree } 3$$

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

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-4,3,0 ; \quad f(2)=-36$$

Sketch the graph of \(f\) for the indicated value of \(c\) or \(a\) $$f(x)=a x^{3}+2$$ (a) \(a=2\) b) \(a=-\frac{1}{3}\)

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=3 x^{3}+2 x-4 ; \quad p(x)=2 x^{2}+1$$

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 \(\mathbb{R}\). $$-3,1-7 i; \quad \text { degree } 3$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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. $$f(x)=\frac{2(x+4)(x+2)}{5(x+2)(x-1)}$$

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-3,-2,0 ; \quad f(-4)=16$$

Sketch the graph of \(f\) for the indicated value of \(c\) or \(a\) $$f(x)=a x^{3}-3$$ (a) \(a=-2\) b) \(a=\frac{1}{4}\)

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=3 x^{3}-5 x^{2}-4 x-8 ; \quad p(x)=2 x^{2}+x$$

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 \(\mathbb{R}\). $$-1,0,3+i ; \quad \text { degree } 4$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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 \(y=25\)

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-2 i, 2 i, 3 ; \quad f(1)=20$$

Use the Intermediate value theorem to show that \(f\) has a zero between \(a\) and \(b\) $$f(x)=x^{3}-4 x^{2}+3 x-2 ; \quad a=3, \quad b=4$$

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=7 x+2 ; \quad p(x)=2 x^{2}-x-4$$

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 \(\mathbb{R}\). $$0,2,-2-i; \quad \text { degree } 4$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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 \(q=1.4\)

Find a polynomial \(f(x)\) of degree 3 that has the indicated zeros and satisfies the given condition. $$-3 i, 3 i, 4 ; \quad f(-1)=50$$

Use the Intermediate value theorem to show that \(f\) has a zero between \(a\) and \(b\) $$f(x)=2 x^{3}+5 x^{2}-3 ; \quad a=-3, \quad b=-2$$

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=-5 x^{2}+3 ; \quad p(x)=x^{3}-3 x+9$$

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 \(\mathbb{R}\). $$4+3 i,-2+i ; \text { degree } 4$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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 \(f\) $$f(x)=\frac{3}{x-4}$$

Find a polynomial \(f(x)\) of degree 4 with leading coefficient I such that both \(-4\) and 3 are zeros of multiplicity 2 and sketch the graph of \(f .\)

Use the Intermediate value theorem to show that \(f\) has a zero between \(a\) and \(b\) $$f(x)=-x^{4}+3 x^{3}-2 x+1 ; \quad a=2, \quad b=3$$

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

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 \(\mathbb{R}\). $$3+5 i,-1-i ; \text { degree } 4$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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 \(r=40\)

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

Find a polynomial \(f(x)\) of degree 4 with leading coefficient 1 such that both \(-5\) and 2 are zeros of multiplicity 2 and sketch the graph of \(f\).

Use the Intermediate value theorem to show that \(f\) has a zero between \(a\) and \(b\) $$f(x)=2 x^{4}+3 x-2 ; \quad a=\frac{1}{2}, \quad b=\frac{3}{4}$$

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

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 \(\mathbb{R}\). $$0,-2 i, 1-i ; \quad$ \text { degree }5$$

Exer. 1-12: Express the statement as a formula that involves the given variables and a constant of proportionallty \(k,\) 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 \(y=16\)

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

Find a polynomial \(f(x)\) of degree 6 such that 0 and 3 are both zeros of multiplicity 3 and \(f(2)=-24 .\) Sketch the graph of \(f\).

Use the Intermediate value theorem to show that \(f\) has a zero between \(a\) and \(b\) $$f(x)=x^{5}+x^{3}+x^{2}+x+1 ; \quad a=-\frac{1}{2}, \quad b=-1$$