Chapter 3

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

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. $$ f(x)=x^{3}-2 x^{2}-11 x+12 $$

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. \(f(x)=x^{4}+6 x^{3}-18 x^{2} ;\) between 2 and 3

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

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=-3(x-2)^{2}+12\)

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(s\) varies jointly as \(g\) and the square of \(t\)

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. $$ f(x)=2 x^{3}-3 x^{2}-11 x+6 $$

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. \(f(x)=x^{3}+x^{2}-2 x+1 ;\) between \(-3\) and \(-2\)

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

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=-2(x+1)^{2}+5\)

Write an equation that expresses each relationship. Use \(k\) as the constant of variation. \(V\) varies jointly as \(h\) and the square of \(r\)

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. $$ f(x)=2 x^{3}-5 x^{2}+x+2 $$

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. \(f(x)=x^{5}-x^{3}-1 ;\) between 1 and 2

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{x^{4}-81}{x-3}$$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=-2(x+4)^{2}-8\)

Determine the constant of variation for each stated condition. \(y\) varies directly as \(x,\) and \(y=75\) when \(x=3\)

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. $$ f(x)=3 x^{3}+7 x^{2}-22 x-8 $$

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. \(f(x)=3 x^{3}-10 x+9 ;\) between \(-3\) and \(-2\)

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{6 x^{3}+13 x^{2}-11 x-15}{3 x^{2}-x-3}$$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=2 x^{2}-8 x+3\)

Determine the constant of variation for each stated condition. \(y\) varies directly as \(x,\) and \(y=55\) when \(x=11\)

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. $$ f(x)=3 x^{3}+8 x^{2}-15 x+4 $$

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. \(f(x)=3 x^{3}-8 x^{2}+x+2 ;\) between 2 and 3

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{x^{4}+2 x^{3}-4 x^{2}-5 x-6}{x^{2}+x-2}$$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=3 x^{2}-12 x+1\)

Determine the constant of variation for each stated condition. \(y\) varies directly as \(x^{2},\) and \(y=45\) when \(x=3\)

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

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. $$ x^{3}-2 x^{2}-11 x+12=0 $$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{18 x^{4}+9 x^{3}+3 x^{2}}{3 x^{2}+1}$$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=-x^{2}-2 x+8\)

Determine the constant of variation for each stated condition. \(y\) varies directly as \(x^{2},\) and \(y=72\) when \(x=6\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$x^{4}+13 x^{2}+36=0 ; 3 i$$

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. $$x^{3}-2 x^{2}-7 x-4=0$$

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{2 x^{5}-8 x^{4}+2 x^{3}+x^{2}}{2 x^{3}+1}$$

Find the coordinates of the vertex for the parabola defined by the given quadratic function. \(f(x)=-2 x^{2}+8 x-1\)

Determine the constant of variation for each stated condition. \(W\) varies inversely as \(r,\) and \(W=500\) when \(r=10\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ 3 x^{3}-7 x^{2}+8 x-2=0 ; 1+i $$

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. $$ x^{3}-10 x-12=0 $$

Divide using synthetic division. $$\left(2 x^{2}+x-10\right) \div(x-2)$$

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. \(f(x)=(x-4)^{2}-1\)

Determine the constant of variation for each stated condition. \(T\) varies inversely as \(n,\) and \(T=7\) when \(n=12\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ x^{3}-7 x^{2}+16 x-10=0 ; 3+i $$

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. $$ x^{3}-5 x^{2}+17 x-13=0 $$

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

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. \(f(x)=(x-1)^{2}-2\)

Determine the constant of variation for each stated condition. \(A\) varies directly as \(B\) and inversely as \(C,\) and \(A=9\) when \(B=12\) and \(C=4\)

In Exercises \(15-22,\) use the given root to find the solution set of the polynomial equation. $$ x^{4}-6 x^{2}+25=0 ; 2-i $$

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. $$ 6 x^{3}+25 x^{2}-24 x+5=0 $$

Divide using synthetic division. $$\left(3 x^{2}+7 x-20\right) \div(x+5)$$

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. \(f(x)=(x-1)^{2}+2\)