Polynomials and Polynomials Functions

In the following exercises, find the product of $(x^5 + y^5 )(x^5 - y^5 )$.

Use synthetic division to find the quotient and remainder.

$x^3-6x^2+5x+14$ is divided by $x+1$

Add or subtract the polynomials.

$4p+11p$

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial $\text{ (a) }-5\text{ (b) }8y^3-7y^2-y-3\text{ (c) }-3x^{2}y-5xy+9x{y}^3\text{ (d) }81m^2-4n^2\text{ (e) }-3x^6{y}^{3}z$

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial

$\text{ (a) }64k^3-8\text{ (b) }9m^3+4m^2-2\text{ (c) }\frac{5}{6}\text{ (d) }8a^4-7a^{3}b-6a^2{b}^2-4a{b}^3+7b^4\text{ (e) }-p^4{q}^3$

Add or subtract: $\text{ (a) }12q^2+9q^2\text{ (b) }8m{n}^3-\left(-5m{n}^3\right)\text{. }$

Add or subtract: $\text{ (a) }-15c^2+8c^2\text{ (b) }-15y^2{z}^3-\left(-5y^2{z}^3\right)$

$\text{ Add: (a) }8y^2+3z^2-3y^2\text{ (b) }{m}^2{n}^2-8m^2+4n^2\text{. }$

$\text{ Add: (a) }3m^2+n^2-7m^2\text{ (b) }p{q}^2-6p-5q^2\text{. }$

$\text{ Find the sum: }\left(7x^2-4x+5\right)+\left(x^2-7x+3\right)$

$\text{ Find the sum: }\left(14y^2+6y-4\right)+\left(3y^2+8y+5\right)$

$\text{ Find the difference: }\left(8x^2+3x-19\right)-\left(7x^2-14\right)$

$\text{ Find the}\text{:difference }\left(9b^2-5b-4\right)-\left(3b^2-5b-7\right)$

$\text{ Subtract }\left(a^2+5ab-6b^2\right)\text{ from }\left(a^2+b^2\right)$

$\text{ Subtract }\left(m^2-7mn-3n^2\right)\text{ from }\left(m^2+n^2\right)$

$\text{ Find the sum: }\left(3x^2-4xy+5y^2\right)+\left(2x^2-xy\right)$

$\text{ Find the sum: }\left(2x^2-3xy-2y^2\right)+\left(5x^2-3xy\right)$

$\text{ Simplify: }\left(x^3-x^{2}y\right)-\left(x{y}^2+y^3\right)+\left(x^{2}y+x{y}^2\right)$

$\text{ Simplify: }\left(p^3-p^{2}q\right)+\left(p{q}^2+q^3\right)-\left(p^{2}q+p{q}^2\right)$

$\text{ For the function }f\left(x\right)=3x^2+2x-15,\text{ find (a) }f\left(3\right)\text{ (b) }f(-5)\text{ (c) }f\left(0\right)\text{. }$

$\text{ For the function }g\left(x\right)=5x^2-x-4,\text{ find (a) }g(-2)\text{ (b) }g(-1)\text{ (c) }g\left(0\right)\text{. }$

The polynomial function $h\left(t\right)=-16t^2+150$ gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after t = 0 seconds (the initial height of the object). 

The polynomial function $h\left(t\right)=-16t^2+175$ gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after t = 3 seconds. 

For functions$f\left(x\right)=2x^2-4x+3$ and $g\left(x\right)=x^2-2x-6$ find: $\left(a\right)(f+g)\left(x\right)\left(b\right)(f+g)\left(3\right)\left(c\right)(f-g)\left(x\right)\left(d\right)(f-g)(-2)$

$\text{For functions}f\left(x\right)=5x^2-4x-1\text{ and }g\left(x\right)=x^2+3x+8,\text{find }\left(a\right)(f+g)\left(x\right)\left(b\right)(f+g)\left(3\right)\left(c\right)(f-g)\left(x\right)\left(d\right)(f-g)(-2).$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. 

$\begin{array}{l}\left(a\right)47x^5-17x^2{y}^3+y^2\\ \left(b\right)5c^3+11c^2-c-8\\ \left(c\right)\frac{5}{9}ab+\frac{1}{3}b\\ \left(d\right)4\\ \left(e\right)4pq+17\end{array}$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. 

$\begin{array}{l}\left(a\right)x^2-y^2\\ \left(b\right)-13c^4\\ \left(c\right)a^2+2ab-7b^2\\ \left(d\right)4x^2{y}^2-3xy+8\\ \left(e\right)19\end{array}$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial.

$$\begin{gathered} \left(a\right)8y-5x \\ \left(b\right)y^2-5yz-6z^2 \\ \left(c\right)y^3-8y^2+2y-16 \\ \left(d\right)81a{b}^4-24a^2{b}^2+3b \\ \left(e\right)-18 \end{gathered}$$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. 

$\begin{array}{l}\left(a\right)11y^2\\ \left(b\right)-73\\ \left(c\right)6x^2-3xy+4x-2y+y^2\\ \left(d\right)4y^2+17z^2\\ \left(e\right)5c^3+11c^2-c-8\end{array}$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. 

$\begin{array}{l}\left(a\right)5a^2+12ab-7b^2\\ \left(b\right)18x{y}^{2}z\\ \left(c\right)5x+2\\ \left(d\right)y^3-8y^2+2y-16\\ \left(e\right)-24\end{array}$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. 

$$\begin{gathered} \left(a\right)9y^3-10y^2+2y-6 \\ \left(b\right)-12p^{3}q \\ \left(c\right)a^2+9ab+18b^2 \\ \left(d\right)20x^2{y}^2-10a^2{b}^2+30 \\ \left(e\right)17 \end{gathered}$$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. 

$\begin{array}{l}\left(a\right)14s-29t\\ \left(b\right)z^2-5z-6\\ \left(c\right)y^3-8y^{2}z+2y{z}^2-16z^3\\ \left(d\right)23a{b}^2-14\\ \left(e\right)-3\end{array}$

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial

$\begin{array}{l}\left(a\right)15xy\\ \left(b\right)15\\ \left(c\right)6x^2-3xy+4x-2y+y^2\\ \left(d\right)10p-9q\\ \left(e\right)m^4+4m^3+6m^2+4m+1\end{array}$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\left(a\right)7x^2+5x^2\\ \left(b\right)4a-9a\end{array}$

In the following exercises, add or subtract the monomials 

$$\begin{gathered} \left(a\right)4y^3+6y^3 \\ \left(b\right)-y-5y \end{gathered}$$

In the following exercises, add or subtract the monomials.

$\begin{array}{l}\left(a\right)-12w+18w\\ \left(b\right)7x^{2}y-\left(-12x^{2}y\right)\end{array}$

In the following exercises, add or subtract the monomials. 

$$\begin{gathered} \left(a\right)-3m+9m \\ \left(b\right)15y{z}^2-\left(-8y{z}^2\right) \end{gathered}$$

In the following exercises, add or subtract the monomials.

$7x^2+5x^2+4a-9a$

In the following exercises, add or subtract the monomials. 

$4y^3+6y^3-y-5y$

In the following exercises, add or subtract the monomials. 

$-12w+18w+7x^{2}y-\left(-12x^{2}y\right)$

In the following exercises, add or subtract the monomials. 

$-3m+9m+15y{z}^2-\left(-8y{z}^2\right)$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\text{(a) }-5b-17b\\ \text{(b) }3xy-(-8xy)+5xy\end{array}$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\left(a\right)-10x-35x\\ \left(b\right)17m{n}^2-\left(-9m{n}^2\right)+3m{n}^2\end{array}$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\left(a\right)12a+5b-22a\\ \left(b\right)p{q}^2-4p-3q^2\end{array}$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\left(a\right)14x-3y-13x\\ \left(b\right)a^{2}b-4a-5a{b}^2\end{array}$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\left(a\right)2a^2+b^2-6a^2\\ \left(b\right)x^{2}y-3x+7x{y}^2\end{array}$

In the following exercises, add or subtract the monomials. 

$$\begin{gathered} \left(a\right)5u^2+4v^2-6u^2 \\ \left(b\right)12a+8b \end{gathered}$$

In the following exercises, add or subtract the monomials. 

$\begin{array}{l}\left(a\right)x{y}^2-5x-5y^2\\ \left(b\right)19y+5z\end{array}$

In the following exercises, add or subtract the monomials. 

$12a+5b-22a+p{q}^2-4p-3q^2$

In the following exercises, add or subtract the monomials. 

$14x-3y-13x+a^{2}b-4a-5a{b}^2$