Chapter 12

Write the polynomials in standard form. $$ 3 x-2 x^{2}+5 x^{7}+4 x^{5} $$

Find the zeros of the polynomials. $$ (x-3)(x-4)(x+2) $$

Which of the expressions in Exercises \(1-6\) are equivalent to monomials in \(x ?\) $$ -\frac{x^{3}}{5} $$

Write the polynomials in standard form. $$ 3 x^{2}+2 x+2 x^{7}-5 x^{2}-3 x^{7} $$

Find the zeros of the polynomials. $$ x(x+5)(x-7)^{2} $$

Write the polynomials in standard form. $$ x(x-2)+x^{2}(3-x) $$

Find the zeros of the polynomials. $$ x^{2}-x-6 $$

Which of the expressions are equivalent to monomials in \(x ?\) $$ -x \cdot x^{2} $$

Find the zeros of the polynomials. $$ x^{4}-4 x^{3}+4 x^{2} $$

Write the polynomials in standard form. $$ \frac{x^{4}-2 x-14 x^{3}}{7} $$

Give the constant term, \(a_{0}\). $$ 4 t^{3}-2 t^{2}+17 $$

Find the zeros of the polynomials. $$ x^{2}+1 $$

Give the constant term, \(a_{0}\). $$ 12 t-2 t^{3}+6 $$

Find the zeros of the polynomials. $$ x^{4}-1 $$

Give all the solutions of the equations. $$ (x-1)(x+2)(x-3)=0 $$

Give the constant term, \(a_{0}\). $$ 15-11 t^{9}-8 t^{4} $$

Which of the expressions in Exercises \(7-12\) are polynomials in \(x ?\) If an expression is not a polynomial in \(x,\) what rules it out? $$ \frac{2 x^{3}}{5}-2 x^{7} $$

Find possible formulas for the polynomial functions described. The graph crosses the \(x\) -axis at \(x=-2\) and \(x=3\) and its long-run behavior is like \(y=-2 x^{2}\).

Give all the solutions of the equations. $$ (x+3)\left(1-x^{2}\right)=0 $$

Give the constant term, \(a_{0}\). $$ 7 t^{3}+2 t^{2}+5 t $$

Which of the expressions are polynomials in \(x ?\) If an expression is not a polynomial in \(x,\) what rules it out? $$ x(x-1)-x^{2}\left(1-x^{3}\right) $$

Find possible formulas for the polynomial functions described. The graph bounces off the \(x\) -axis at \(x=-2\), crosses the \(x\) -axis at \(x=3\), and has long-run behavior like \(y=x^{3} .\)

Give all the solutions of the equations. $$ x^{3}+3 x^{2}+2 x=0 $$

Give the constant term, \(a_{0}\). $$ (3 t+1)(2 t-1) $$

Which of the expressions are polynomials in \(x ?\) If an expression is not a polynomial in \(x,\) what rules it out? $$ \sqrt{2} x-x^{2}+x^{4} $$

Find possible formulas for the polynomial functions described. The graph bounces off the \(x\) -axis at \(x=-2\), crosses the \(x\) -axis at \(x=3\), and has long-run behavior like \(y=x^{5}\)

Give all the solutions of the equations. $$ x^{3}-2 x^{2}+2^{2} x-2^{3}=0 $$

Give the constant term, \(a_{0}\). $$ t(t-1)(t-2) $$

Which of the expressions are polynomials in \(x ?\) If an expression is not a polynomial in \(x,\) what rules it out? $$ \sqrt{2 x}-x^{3}+x^{5} $$

The polynomial \(p(x)\) can be written in two forms: I. \(\quad p(x)=2 x^{3}-3 x^{2}-11 x+6\) II. \(p(x)=(x-3)(x+2)(2 x-1)\) Which form most readily shows (a) The zeros of \(p(x) ?\) What are they? (b) The vertical intercept? What is it? (c) The sign of \(p(x)\) as \(x\) gets large, either positive or negative? What are the signs? (d) The number of times \(p(x)\) changes sign as \(x\) increases from large negative to large positive \(x ?\) How many times is this?

Give all the solutions of the equations. $$ (x-1) x(x+3)=0 $$

Find the degree. $$ 2 s^{6}-3 s^{5}-6 s^{4}-4 s+1 $$

A polynomial \(p(x)\) can be written in two forms: I. \(p(x)=\left(x^{2}+4\right)\left(4-x^{2}\right)\) II. \(\quad p(x)=16-x^{4}\) Which form most readily shows (a) The number of zeros of \(p(x) ?\) Find them. (b) The vertical intercept? What is it? (c) The sign of \(p(x)\) as \(x\) gets large, either positive or negative. What are the signs?

Give all the solutions of the equations. $$ x^{4}+x^{2}-2=0 $$

Find the degree. $$ 2 s^{3}-s^{2}+1-s^{3}+2 s^{2}-s+3 s^{3} $$

Graph \(y=2(x-3)^{2}(x+1)\). Label all axis intercepts.

Give all the solutions of the equations. $$ s\left(s^{2}+1\right)=s\left(2 s^{2}-3\right) $$

Find the degree. $$ 3 s^{2}+2 s^{4}+s-s^{4}+2 s^{3}-1-s^{4}+3 s $$

In Exercises \(13-18,\) is the expression a polynomial in the given variable? $$ \left(4-2 p^{2}\right) p+3 p-(p+2)^{2}, \text { in } p $$

Give all the solutions of the equations. $$ (t+3)^{3}+4(t+3)^{2}=0 $$

Give the leading term. $$ 3 x^{5}-2 x^{3}+4 $$

Is the expression a polynomial in the given variable? $$ (x-1)(x-2)(x-3)(x-4)+29, \text { in } x $$

Explain why \(p(x)=x^{5}+3 x^{3}+2\) must have at least one zero.

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

Give the leading term. $$ 2 x^{7}-4 x^{11}+6 $$

Is the expression a polynomial in the given variable? $$ \left(a+\frac{1}{x}\right)^{2}-\left(a-\frac{1}{x}\right)^{2}, \text { in } x $$

Find two different formulas for a polynomial \(p(x)\) of degree 3 with \(p(0)=6\) and \(p(-2)=0,\) and graph them.

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

Give the leading term. $$ 12-3 x^{5}-15 x^{3} $$

Is the expression a polynomial in the given variable? $$ \left(a+\frac{1}{x}\right)^{2}-\left(a-\frac{1}{x}\right)^{2}, \text { in } a $$