Chapter 4

If the polynomial function $$ P(x)=a_{e} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{i} x+a_{0} $$ has integer coefficients, then the only numbers that could possibly be rational zeros of \(P\) are all of the form \(\frac{p}{q},\) where \(p\) is a factor of _________ and \(q\) is a factor of _________The possible rational zeros of \(P(x)=6 x^{3}+5 x^{2}-19 x-10\) are _________

The polynomial \(P(x)=3(x-5)^{3}(x-3)(x+2)\) has degree ________. It has zeros \(5,3,\) and _________. The zero 5 has multiplicity ________. and the zero 3 has multiplicity_________.

Using Descartes' Rule of Signs, we can tell that the polynomial \(P(x)=x^{5}-3 x^{4}+2 x^{3}-x^{2}+8 x-8\) has _________, _________, or __________positive real zeros and _________ negative real zeros.

If the rational function \(y=r(x)\) has the horizontal asymptote \(y=2,\) then \(y \rightarrow\) ________ as \(X \rightarrow \pm \infty\)

(a) If a is a zero of the polynomial P, then ________ must be a factor of \(P(x)\). (b) If \(a\) is a zero of multiplicity \(m\) of the polynomial \(P,\) then ________ must be a factor of \(P(x)\) when we factor \(P\) completely

(a) If we divide the polynomial \(P(x)\) by the factor \(x-c\) and we obtain a remainder of 0 , then we know that \(c\) is a ________of \(P\)

Every polynomial has one of the following behaviors: $$\begin{array}{l}{\text { (i) } y \rightarrow \infty \text { as } x \rightarrow \infty \text { and } y \rightarrow \infty \text { as } x \rightarrow-\infty} \\ {\text { (ii) } y \rightarrow \infty \text { as } x \rightarrow \infty \text { and } y \rightarrow-\infty \text { as } x \rightarrow-\infty} \\ {\text { (iii) } y \rightarrow-\infty \text { as } x \rightarrow \infty \text { and } y \rightarrow \infty \text { as } x \rightarrow-\infty} \\ {\text { (iv) } y \rightarrow-\infty \text { as } x \rightarrow \infty \text { and } y \rightarrow-\infty \text { as } x \rightarrow-\infty}\end{array}$$ For each polynomial, choose the appropriate description of its end behavior from the list above. (a) \(y=x^{3}-8 x^{2}+2 x-15 :\) end behavior _____. (b) \(y=-2 x^{4}+12 x+100 :\) end behavior _____.

True or false? If \(c\) is a real zero of the polynomial \(P,\) then all the other zeros of \(P\) are zeros of \(P(x) /(x-c) .\)

\(3-8=\) Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\) $$ P(x)=3 x^{2}+5 x-4, \quad D(x)=x+3 $$

A polynomial of degree \(n \geq 1\) has exactly ________ zeros if a zero of multiplicity m is counted m times.

If \(c\) is a zero of the polynomial \(P,\) which of the following statements must be true? (a) \(P(c)=0\) (b) \(P(0)=c\) (c) \(x-c\) is a factor of \(P(x)\) . (d) \(c\) is the \(y\) -intercept of the graph of \(P\) .

The graph of \(f(x)=2(x-3)^{2}+5\) is a parabola that opens ________, with its vertex at ( _____, ____) and \(f(3)=\) _________ is the (minimum/maximum) _________ value of \(f\)

True or false? If \(a\) is an upper bound for the real zeros of the polynomial \(P\) , then \(-a\) is necessarily a lower bound for the real zeros of \(P\) .

The following questions are about the rational function $$ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function r has y-intercept ____________

\(3-8=\) Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\) $$ P(x)=x^{3}+4 x^{2}-6 x+1, \quad D(x)=x-1 $$

If the polynomial function \(P\) has real coefficients and if \(a+b i\) is a zero of \(P,\) then _________ is also a zero of \(P\).

Which of the following statements couldn't possibly be true about the polynomial function \(P ?\) (a) \(P\) has degree \(3,\) two local maxima, and two local minima. (b) \(P\) has degree 3 and no local maxima or minima. (c) \(P\) has degree \(4,\) one local maximum, and no local minima.

List all possible rational zeros given by the Rational Zeros Theorem (but don' t check to see which actually are zeros). $$ P(x)=x^{3}-4 x^{2}+3 $$

The following questions are about the rational function $$ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has vertical asymptotes \(x=\) ________ and \(\boldsymbol{X}=\) ______

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

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

The graph of a quadratic function is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. $$ f(x)=-x^{2}+6 x-5 $$

List all possible rational zeros given by the Rational Zeros Theorem (but don' t check to see which actually are zeros). $$ Q(x)=x^{4}-3 x^{3}-6 x+8 $$

The following questions are about the rational function $$ I(x)=\frac{(x+1)(x-2)}{(x+2)(x-3)} $$ The function \(r\) has horizontal asymptote \(y=\) ___________

\(3-8=\) Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\) $$P(x)=4 x^{3}+7 x+9, \quad D(x)=2 x+1$$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{5}+9 x^{3}\)

The graph of a quadratic function is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. $$ f(x)=-x^{2}+6 x-5 $$

List all possible rational zeros given by the Rational Zeros Theorem (but don' t check to see which actually are zeros). $$ R(x)=2 x^{5}+3 x^{3}+4 x^{2}-8 $$

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

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{3}-2 x^{2}+2 x\)

\(5-8\) Sketch the graph of each function by transforming the graph of an appropriate function of the form \(y=x^{n}\) from Figure 2 . Indicate all \(x\) - and \(y\) -intercepts on each graph. $$ \begin{array}{ll}{\text { (a) } P(x)=x^{3}-8} & {\text { (b) } Q(x)=-x^{3}+27} \\\ {\text { (c) } R(x)=-(x+2)^{3}} & {\text { (d) } S(x)=\frac{1}{2}(x-1)^{3}+4}\end{array} $$

The graph of a quadratic function is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. $$ f(x)=2 x^{2}-4 x-1 $$

List all possible rational zeros given by the Rational Zeros Theorem (but don' t check to see which actually are zeros). $$ S(x)=6 x^{4}-x^{2}+2 x+12 $$

A rational function is given. (a) Complete each table for the function. (b) Describe the behavior of the function near its vertical asymptote, based on Tables 1 and 2. (c) Determine the horizontal asymptote, based on Tables 3 and 4. $$ r(x)=\frac{4 x+1}{x-2} $$

\(3-8=\) Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\) $$P(x)=2 x^{5}+4 x^{4}-4 x^{3}-x-3, \quad D(x)=x^{2}-2$$

A polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{3}+x^{2}+x\)

The graph of a quadratic function is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. $$ f(x)=3 x^{2}+6 x-1 $$

List all possible rational zeros given by the Rational Zeros Theorem (but don' t check to see which actually are zeros). $$ T(x)=4 x^{4}-2 x^{2}-7 $$

\(9-14\) . 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 polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{4}+2 x^{2}+1\)

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

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

\(9-14\) . 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 polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{4}-x^{2}-2\)

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. $$ f(x)=x^{2}+8 x $$

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

\(9-14\) . 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 polynomial P is given. (a) Find all zeros of P, real and complex. (b) Factor P completely. \(P(x)=x^{4}-16\)

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. $$ s(x)=\frac{3 x}{x-5} $$