Chapter 5

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{3}-15 x^{2}+75 x-125\right) \div(x-5) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=x^{5}-2 x,\) between \(x=1\) and \(x=2\).

For the following exercises, find the intercepts of the functions. $$ g(n)=-2(3 n-1)(2 n+1) $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(2,0),(x, y)=(4,4) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the square root of \(x\). When \(x=16,\) then \(y=4 .\) Find \(y\) when \(x=36\)

For the following exercises, find the inverse of the functions. $$ f(x)=\frac{5 x+1}{2-5 x} $$

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{-2 x}{x-6} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{3}-3 x^{2}-32 x-15=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(9 x^{3}-x+2\right) \div(3 x-1) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=-x^{4}+4,\) between \(x=1\) and \(x=3\).

For the following exercises, find the intercepts of the functions. $$ f(x)=x^{4}-16 $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(-2,-1),(x, y)=(-4,3) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies directly as the cube root of \(x\). When \(x=125,\) then \(y=15\). Find \(y\) when \(x=1,000\).

For the following exercises, find the inverse of the functions. $$ f(x)=x^{2}+2 x,[-1, \infty) $$

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{x^{2}-4 x+3}{x^{2}-4 x-5} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{3}+x^{2}-7 x-6=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(6 x^{3}-x^{2}+5 x+2\right) \div(3 x+1) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=-2 x^{3}-x,\) between \(x=-1\) and \(x=1\).

For the following exercises, find the intercepts of the functions. $$ f(x)=x^{3}+27 $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(0,1),(x, y)=(2,5) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with \(x\). When \(x=3\), then \(y=2\). Find \(y\) when \(x=1\).

For the following exercises, find the inverse of the functions. $$ f(x)=x^{2}+4 x+1,[-2, \infty) $$

For the following exercises, describe the local and end behavior of the functions. $$ f(x)=\frac{2 x^{2}-32}{6 x^{2}+13 x-5} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{3}-3 x^{2}-x+1=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}+x^{3}-3 x^{2}-2 x+1\right) \div(x+1) $$

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. \(f(x)=x^{3}-100 x+2\), between\(x=0.01\) and \(x=0.1\)

For the following exercises, find the intercepts of the functions. $$ f(x)=x\left(x^{2}-2 x-8\right) $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(2,3),(x, y)=(5,12) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the square of \(x\). When \(x=4,\) then \(y=3\). Find \(y\) when \(x=2\).

For the following exercises, find the inverse of the functions. $$ f(x)=x^{2}-6 x+3,[3, \infty) $$

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{24 x^{2}+6 x}{2 x+1} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 3 x^{3}-x^{2}-11 x-6=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}-3 x^{2}+1\right) \div(x-1) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=(x+2)^{3}(x-3)^{2} $$

For the following exercises, find the intercepts of the functions. $$ f(x)=(x+3)\left(4 x^{2}-1\right) $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(-5,3),(x, y)=(2,9) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the cube of \(x\). When \(x=3\), then \(y=1\). Find \(y\) when \(x=1\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=x^{2}+2, \quad x \geq 0 $$

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{4 x^{2}-10}{2 x-4} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{3}-5 x^{2}+9 x-9=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}+2 x^{3}-3 x^{2}+2 x+6\right) \div(x+3) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x^{2}(2 x+3)^{5}(x-4)^{2} $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(3,2),(x, y)=(10,1) $$

For the following exercises, use the given information to find the unknown value. \(y\) varies inversely with the square root of \(x .\) When \(x=64,\) then \(y=12 .\) Find \(y\) when \(x=36\).

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=4-x^{2}, \quad x \geq 0 $$

For the following exercises, find the slant asymptote of the functions. $$ f(x)=\frac{81 x^{2}-18}{3 x-2} $$

For the following exercises, use the Rational Zero Theorem to find the real solution(s) to each equation. $$ 2 x^{3}-3 x^{2}+4 x+3=0 $$

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}-10 x^{3}+37 x^{2}-60 x+36\right) \div(x-2) $$

For the following exercises, find the zeros and give the multiplicity of each. $$ f(x)=x^{3}(x-1)^{3}(x+2) $$

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(0,1),(x, y)=(1,0) $$