Chapter 6

State the number of positive real zeros, negative real zeros, and imaginary zeros for each function. \(h(x)=4 x^{3}-6 x^{2}+8 x-5\)

Factor completely. If the polynomial is not factorable, write prime. $$ 3 a x-15 a+x-5 $$

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=x^{2}-3 x+8\)

Simplify. $$ \left(b^{3}+8 b^{2}-20 b\right) \div(b-2) $$

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ 5 x^{2} y^{4}+x \sqrt{3} $$

Simplify. Assume that no variable equals 0. $$ \left(n^{4}\right)^{4} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}+x^{2}-16 x-16 ; x+4 $$

Find all of the rational zeros of each function. \(g(x)=x^{4}-3 x^{3}+x^{2}-3 x\)

State the number of positive real zeros, negative real zeros, and imaginary zeros for each function. \(q(x)=x^{4}+5 x^{3}+2 x^{2}-7 x-9\)

Factor completely. If the polynomial is not factorable, write prime. $$ y^{2}-5 y+4 $$

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=2 x^{3}-x^{2}+5 x-7\)

Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial. $$ \frac{4}{3} y^{2}+\frac{5}{6} y^{7} $$

Simplify. $$ \left(g^{2}+8 g+15\right)(g+3)^{-1} $$

Simplify. Assume that no variable equals 0. $$ \left(z^{2}\right)^{5} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}-6 x^{2}+11 x-6 ; x-2 $$

Find all of the rational zeros of each function. \(p(x)=x^{4}+10 x^{3}+33 x^{2}+38 x+8\)

Factor completely. If the polynomial is not factorable, write prime. $$ 2 b^{2}+13 b-7 $$

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=x^{5}-x^{2}\)

Simplify. $$ \left(3 x^{2}-x+2\right)+\left(x^{2}+4 x-9\right) $$

Simplify. $$ \frac{y^{3}+3 y^{2}-5 y-4}{y+4} $$

Simplify. Assume that no variable equals 0. $$ (2 x)^{4} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 2 x^{3}-5 x^{2}-28 x+15 ; x-5 $$

Find all of the zeros of each function. \(p(x)=6 x^{4}+22 x^{3}+11 x^{2}-38 x-40\)

State the number of positive real zeros, negative real zeros, and imaginary zeros for each function. \(f(x)=x^{10}-x^{8}+x^{6}-x^{4}+x^{2}-1\)

Factor completely. If the polynomial is not factorable, write prime. $$ z^{3}+125 $$

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. \(r(3 a)\)

Simplify. $$ \left(5 y+3 y^{2}\right)+\left(-8 y-6 y^{2}\right) $$

Simplify. $$ \frac{m^{3}+3 m^{2}-7 m-21}{m+3} $$

Simplify. Assume that no variable equals 0. $$ (-2 c)^{3} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 3 x^{3}+10 x^{2}-x-12 ; x+3 $$

Find all of the zeros of each function. \(g(x)=5 x^{4}-29 x^{3}+55 x^{2}-28 x\)

Find all of the zeros of each function. \(g(x)=x^{3}+6 x^{2}+21 x+26\)

Factor completely. If the polynomial is not factorable, write prime. $$ t^{3}-8 $$

Simplify. $$ \left(9 r^{2}+6 r+16\right)-\left(8 r^{2}+7 r+10\right) $$

Simplify. $$ \left(t^{5}-3 t^{2}-20\right)(t-2)^{-1} $$

Simplify. Assume that no variable equals 0. $$ \left(a^{3} b^{3}\right)(a b)^{-2} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 2 x^{3}+7 x^{2}-53 x-28 ; 2 x+1 $$

Find all of the zeros of each function. \(h(x)=6 x^{3}+11 x^{2}-3 x-2\)

Find all of the zeros of each function. \(h(x)=x^{3}-6 x^{2}+10 x-8\)

Write each expression in quadratic form, if possible. $$ 2 x^{4}+6 x^{2}-10 $$

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. \(p\left(\mathrm{a}^{2}\right)\)

Simplify. $$ \left(7 m^{2}+5 m-9\right)+\left(3 m^{2}-6\right) $$

Simplify. $$ \left(y^{5}+32\right)(y+2)^{-1} $$

Simplify. Assume that no variable equals 0. $$ \left(-2 r^{2} s\right)^{3}\left(3 r s^{2}\right) $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 2 x^{3}+17 x^{2}+23 x-42 ; 2 x+7 $$

Find all of the zeros of each function. \(p(x)=x^{3}+3 x^{2}-25 x+21\)

Find all of the zeros of each function. \(f(x)=x^{3}-5 x^{2}-7 x+51\)

Write each expression in quadratic form, if possible. $$ a^{8}+10 a^{2}-16 $$

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. \(p\left(2 a^{3}\right)\)

Simplify. $$ 4 b(c b-z d) $$