Chapter 12

Give all the solutions of the equations. $$ (u+3)^{3}=-(u+3)^{2} $$

Give the leading term. $$ 12 x^{13}+4 x^{5}-11 x^{13} $$

Is the expression a polynomial in the given variable? $$ \frac{n(n+1)(n+2)}{6}, \text { in } n $$

Give all the solutions of the equations. $$ (s+10)^{2}-6(s+10)-16=0 $$

Give the leading term. $$ 13 x^{4}\left(2 x^{2}+1\right) $$

Is the expression a polynomial in the given variable? $$ P\left(1+\frac{r}{12}\right)^{10}, \text { in } r $$

Find possible formulas for the polynomials described. The degree is \(n=2\) and the zeros are \(x=2,-3\).

Give the leading term. $$ x^{8} $$

In Exercises \(19-25, p(z)=4 z^{3}-z\). Find the given values and simplify if possible. $$ p(0) $$

Find possible formulas for the polynomials described. The degree is 5 and the zeros are \(x=-4,-1,0,3,9\).

Give the leading coefficient. $$ 5 x^{6}-4 x^{5}+3 x^{4}-2 x^{3}+x^{2}+1 $$

p(z)=4 z^{3}-z. Find the given values and simplify if possible. $$ p(\sqrt{5}) $$

Find possible formulas for the polynomials described. The degree is \(n=3\) and there is one zero at \(x=5\) and one double zero at \(x=-13\).

Give the leading coefficient. $$ 1-6 r^{2}+40 r-\frac{1}{2} r^{3}+16 r $$

Give the leading coefficient. $$ 100-\sqrt{6} s+15 s^{2} $$

p(z)=4 z^{3}-z. Find the given values and simplify if possible. $$ p(t+1) $$

Give the leading coefficient. $$ \sqrt{7} u^{3}+12 u-4+6 u^{2} $$

Find possible formulas for the polynomials described. The degree is \(n=6\) and there is one simple zero at \(x=-1,\) one double zero at \(x=3\), and one multiple zero at \(x=5\).

Give the leading coefficient. $$ t^{3}-2 t^{2}-\sqrt[3]{9} t^{3}+1 $$

p(z)=4 z^{3}-z. Find the given values and simplify if possible. The values of \(z\) such that \(p(z)=0\)

The profit from selling \(q\) items of a certain product is \(P(q)=36 q-0.0001 q^{3}\) dollars. Find the values of \(q\) such that \(P(q)=0\). Which of these values make sense in the context of the problem? Interpret the values that make sense.

List the nonzero coefficients of the polynomials. $$ 3 u^{4}+6 u^{3}-3 u^{2}+8 u+1 $$

If \(p(x)=x^{4}-2 x^{2}+1,\) find (a) \(p(0)\) (b) \(p(2)\) (c) \(p\left(t^{2}\right)\). (d) The values of \(x\) such that \(p(x)=0\)

American Airlines limits the size of carry-on baggage to 45 linear inches (length \(+\) width \(+\) height \()\), with a weight of no more than 40 pounds. \({ }^{1}\) (a) If the length and width of a piece of luggage both measure \(x\) inches, express the maximum height of the luggage in terms of \(x\). (b) Express the volume of the piece of luggage in part (a) in terms of \(x\). (c) Find the zeros of your equation from part (b). What does this tell you about the dimensions of the piece of luggage?

List the nonzero coefficients of the polynomials. $$ 2 x^{5}-3 x^{3}+x^{7}+1 $$

List the nonzero coefficients of the polynomials. $$ \frac{s^{13}}{3} $$

Find a polynomial of degree 4 that has zeros at \(x=\) \(-2, x=-1, x=2,\) and \(x=3\) and whose graph contains the point (0,6) .

Problems \(28-31\) refer to the functions \(f(x)\) and \(g(x),\) where the function $$ g(x)=1+\frac{1}{2} x+\frac{3}{8} x^{2}+\frac{5}{16} x^{3} $$ is used to approximate the values of $$ f(x)=\frac{1}{\sqrt{1-x}} $$ Evaluate \(f\) and \(g\) at \(x=0\). What does this tell you about the graphs of these two functions?

Find two different polynomials of degree 3 with zeros \(1,2,\) and \(3 .\)

List the nonzero coefficients of the polynomials. $$ \pi x $$

Refer to the functions \(f(x)\) and \(g(x),\) where the function $$ g(x)=1+\frac{1}{2} x+\frac{3}{8} x^{2}+\frac{5}{16} x^{3} $$ is used to approximate the values of $$ f(x)=\frac{1}{\sqrt{1-x}} $$ Evaluate \(f(x)\) and \(g(x)\) at \(x=0.1,0.2,0.3\) and record your answers to three decimal places in a table. Does your table support the claim that \(g(x)\) is a good approximation to \(f(x)\) for these values of \(x\) ?

(a) Find two different polynomials with zeros \(x=-1\) and \(x=5 / 2\). (b) Find a polynomial with zeros \(x=-1\) and \(x=\) \(5 / 2\) and leading coefficient \(4 .\)

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 2 x^{3}+x-2 $$

Refer to the functions \(f(x)\) and \(g(x),\) where the function $$ g(x)=1+\frac{1}{2} x+\frac{3}{8} x^{2}+\frac{5}{16} x^{3} $$ is used to approximate the values of $$ f(x)=\frac{1}{\sqrt{1-x}} $$ Show that \(f(x)\) is undefined at \(x=1\) and \(x=2\), but that \(g(x)\) is defined at these values. Explain why the algebraic operations used to define \(f\) may lead to undefined values, whereas the operations used to define \(g\) will not.

Without solving the equation, decide how many solutions it has. $$ (x-1)(x-2)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 5-3 x^{7} $$

Refer to the functions \(f(x)\) and \(g(x),\) where the function $$ g(x)=1+\frac{1}{2} x+\frac{3}{8} x^{2}+\frac{5}{16} x^{3} $$ is used to approximate the values of $$ f(x)=\frac{1}{\sqrt{1-x}} $$ Given that $$ f(1 / 2)=\frac{1}{\sqrt{1-\frac{1}{2}}}=\frac{1}{\sqrt{\frac{1}{2}}}=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2} $$ use \(g(x)\) to find a rational number (a fraction) that approximately equals \(\sqrt{2}\).

Without solving the equation, decide how many solutions it has. $$ \left(x^{2}+1\right)(x-2)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 20-5 x $$

Without solving the equation, decide how many solutions it has. $$ \left(x^{2}+2 x\right)(x-3)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ \sqrt{7} $$

Without solving the equation, decide how many solutions it has. $$ \left(x^{2}-4\right)(x+5)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ \frac{x^{2} \sqrt[3]{5}}{7} $$

Without solving the equation, decide how many solutions it has. $$ (x-2) x=3(x-2) $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 3-2(x-5)^{2} $$

Without solving the equation, decide how many solutions it has. $$ \left(2-x^{2}\right)(x-4)(5-x)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 15 x-4 x^{3}+12 x-5 x^{4}+9 x-6 x^{5} $$

Without solving the equation, decide how many solutions it has. $$ \left(2+x^{2}\right)(x-4)(5-x)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ (x-3)(2 x-1)(x-2) $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ (x+1)^{3} $$