Chapter 3

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=x^{2}+4 x-8, \quad D(x)=x+3$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=x^{2}-6 x$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{4}-x^{2}-2$$

Find the real and imaginary parts of the complex number. $$-\frac{1}{2}$$

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). $$U(x)=12 x^{5}+6 x^{3}-2 x-8$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=x^{3}+6 x+5, \quad D(x)=x-4$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=x^{2}+8 x$$

Find the \(x\) - and \(y\) -intercepts of the rational function. $$r(x)=\frac{x-1}{x+4}$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{4}-16$$

Find the real and imaginary parts of the complex number. $$-\frac{2}{3} i$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=4 x^{2}-3 x-7, \quad D(x)=2 x-1$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=2 x^{2}+6 x$$

Find the \(x\) - and \(y\) -intercepts of the rational function. $$s(x)=\frac{3 x}{x-5}$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{4}+6 x^{2}+9$$

Find the real and imaginary parts of the complex number. $$i \sqrt{3}$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=6 x^{3}+x^{2}-12 x+5, \quad D(x)=3 x-4$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=-x^{2}+10 x$$

Find the \(x\) - and \(y\) -intercepts of the rational function. $$t(x)=\frac{x^{2}-x-2}{x-6}$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{3}+8$$

Find the real and imaginary parts of the complex number. $$\sqrt{3}+\sqrt{-4}$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=2 x^{4}-x^{3}+9 x^{2}, \quad D(x)=x^{2}+4$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=x^{2}+4 x+3$$

Find the \(x\) - and \(y\) -intercepts of the rational function. $$r(x)=\frac{2}{x^{2}+3 x-4}$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{3}-8$$

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=x^{5}+x^{4}-2 x^{3}+x+1, \quad D(x)=x^{2}+x-1$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=x^{2}-2 x+2$$

Find the \(x\) - and \(y\) -intercepts of the rational function. $$r(x)=\frac{x^{2}-9}{x^{2}}$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{6}-1$$

Evaluate the expression and write the result in the form \(a+b i\) $$(2-5 i)+(3+4 i)$$

Find the quotient and remainder using long division. $$\frac{x^{2}-6 x-8}{x-4}$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=-x^{2}+6 x+4$$

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. (GRAPH CANT COPY) $$P(x)=(x-1)(x+2)$$

Find the \(x\) - and \(y\) -intercepts of the rational function. $$r(x)=\frac{x^{3}+8}{x^{2}+4}$$

A polynomial \(P\) is given. (a) Find all zeros of \(P\), real and complex. (b) Factor \(P\) completely. $$P(x)=x^{6}-7 x^{3}-8$$

Evaluate the expression and write the result in the form \(a+b i\) $$(2+5 i)+(4-6 i)$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{3}-7 x^{2}+14 x-8$$

Find the quotient and remainder using long division. $$\frac{x^{3}-x^{2}-2 x+6}{x-2}$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=-x^{2}-4 x+4$$

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. (GRAPH CANT COPY) $$P(x)=(x-1)(x+1)(x-2)$$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{2}+25$$

Evaluate the expression and write the result in the form \(a+b i\) $$(-6+6 i)+(9-i)$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{3}-3 x-2$$

Find the quotient and remainder using long division. $$\frac{4 x^{3}+2 x^{2}-2 x-3}{2 x+1}$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=2 x^{2}+4 x+3$$

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. (GRAPH CANT COPY) $$P(x)=x(x-3)(x+2)$$

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=4 x^{2}+9$$

Evaluate the expression and write the result in the form \(a+b i\) $$(3-2 i)+\left(-5-\frac{1}{3} i\right)$$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=x^{3}+4 x^{2}-3 x-18$$

Find the quotient and remainder using long division. $$\frac{x^{3}+3 x^{2}+4 x+3}{3 x+6}$$

A quadratic function is given. (a) Express the quadratic function in standard form. (b) Find its vertex and its \(x\) - and \(y\) -intercept(s). (c) Sketch its graph. $$f(x)=-3 x^{2}+6 x-2$$