4.3 Classification of expressions and equations  (Page 2/2)

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$4{x}^{3}{y}^{5}-2x{y}^{3}$ is a binomial of degree 8. The degree of the first term is 8.

$3x+10$ is a binomial of degree 1.

Polynomials of the first degree are called linear polynomials.
Polynomials of the second degree are called quadratic polynomials.
Polynomials of the third degree are called cubic polynomials.
Polynomials of the fourth degree are called fourth degree polynomials.
Polynomials of the $n$ th degree are called $n$ th degree polynomials.
Nonzero constants are polynomials of the 0th degree.

Some examples of these polynomials follow:

$4x-9$ is a linear polynomial.

$3{x}^{2}+5x-7$ is a quadratic polynomial.

$8y-2{x}^{3}$ is a cubic polynomial.

$16{a}^{2}-32{a}^{5}-64$ is a 5th degree polynomial.

${x}^{12}-{y}^{12}$ is a 12th degree polynomial.

$7{x}^{5}{y}^{7}{z}^{3}-2{x}^{4}{y}^{7}z+{x}^{3}{y}^{7}$ is a 15th degree polynomial. The first term is of degree $5+7+3=15$ .

43 is a 0th degree polynomial.

Classification of polynomial equations

As we know, an equation is composed of two algebraic expressions separated by an equal sign. If the two expressions happen to be polynomial expressions, then we can classify the equation according to its degree. Classification of equations by degree is useful since equations of the same degree have the same type of graph. (We will study graphs of equations in Chapter 6.)

The degree of an equation is the degree of the highest degree expression.

Sample set a

$x+7=15$ .

This is a linear equation since it is of degree 1, the degree of the expression on the left of the $"="$ sign.

$5{x}^{2}+2x-7=4$ is a quadratic equation since it is of degree 2.

$9{x}^{3}-8=5{x}^{2}+1$ is a cubic equation since it is of degree 3. The expression on the left of the $"="$ sign is of degree 3.

${y}^{4}-{x}^{4}=0$ is a 4th degree equation.

${a}^{5}-3{a}^{4}=-{a}^{3}+6{a}^{4}-7$ is a 5th degree equation.

$y=\frac{2}{3}x+3$ is a linear equation.

$y=3{x}^{2}-1$ is a quadratic equation.

${x}^{2}{y}^{2}-4=0$ is a 4th degree equation. The degree of ${x}^{2}{y}^{2}-4$ is $2+2=4$ .

Practice set a

Classify the following equations in terms of their degree.

$3x+6=0$

first, or linear

$9{x}^{2}+5x-6=3$

$25{y}^{3}+y=9{y}^{2}-17y+4$

cubic

$x=9$

linear

$y=2x+1$

linear

$3y=9{x}^{2}$

${x}^{2}-9=0$

$y=x$

linear

$5{x}^{7}=3{x}^{5}-2{x}^{8}+11x-9$

eighth degree

Exercises

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term.

$5x+7$

$\begin{array}{lllll}\text{binomial;}\hfill & \hfill & \text{first\hspace{0.17em}(linear);}\hfill & \hfill & 5,7\hfill \end{array}$

$16x+21$

$4{x}^{2}+9$

$\begin{array}{lllll}\text{binomial;}\hfill & \hfill & \text{second\hspace{0.17em}(quadratic);}\hfill & \hfill & 4,9\hfill \end{array}$

$7{y}^{3}+8$

${a}^{4}+1$

$\begin{array}{lllll}\text{binomial;}\hfill & \hfill & \text{fourth;}\hfill & \hfill & 1,1\hfill \end{array}$

$2{b}^{5}-8$

$5x$

$\begin{array}{lllll}\text{monomial;}\hfill & \hfill & \text{first\hspace{0.17em}(linear);}\hfill & \hfill & 5\hfill \end{array}$

$7a$

$5{x}^{3}+2x+3$

$\begin{array}{lllll}\text{trinomial;}\hfill & \hfill & \text{third\hspace{0.17em}(cubic);}\hfill & \hfill & 5\hfill \end{array},2,3$

$17{y}^{4}+{y}^{5}-9$

$41{a}^{3}+22{a}^{2}+a$

$\begin{array}{lllll}\text{trinomial;}\hfill & \hfill & \text{third\hspace{0.17em}(cubic);}\hfill & \hfill & 41\hfill \end{array},22,1$

$6{y}^{2}+9$

$2{c}^{6}+0$

$\begin{array}{lllll}\text{monomial;}\hfill & \hfill & \text{sixth;}\hfill & \hfill & 2\hfill \end{array}$

$8{x}^{2}-0$

$9g$

$\begin{array}{lllll}\text{monomial;}\hfill & \hfill & \text{first\hspace{0.17em}(linear);}\hfill & \hfill & 9\hfill \end{array}$

$5xy+3x$

$3yz-6y+11$

$\begin{array}{lllll}\text{trinomial;}\hfill & \hfill & \text{second\hspace{0.17em}(quadratic);}\hfill & \hfill & 3,-6,11\hfill \end{array}$

$7a{b}^{2}{c}^{2}+2{a}^{2}{b}^{3}{c}^{5}+{a}^{14}$

${x}^{4}{y}^{3}{z}^{2}+9z$

$\begin{array}{lllll}\text{binomial;}\hfill & \hfill & \text{ninth;}\hfill & \hfill & 1,9\hfill \end{array}$

$5{a}^{3}b$

$6+3{x}^{2}{y}^{5}b$

$\begin{array}{lllll}\text{binomial;}\hfill & \hfill & \text{eighth;}\hfill & \hfill & 6,3\hfill \end{array}$

$-9+3{x}^{2}+2xy6{z}^{2}$

5

$\begin{array}{lllll}\text{monomial;}\hfill & \hfill & \text{zero;}\hfill & \hfill & 5\hfill \end{array}$

$3{x}^{2}{y}^{0}{z}^{4}+12{z}^{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}y\ne 0$

$4x{y}^{3}{z}^{5}{w}^{0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}w\ne 0$

$\begin{array}{lllll}\text{monomial;}\hfill & \hfill & \text{ninth;}\hfill & \hfill & 4\hfill \end{array}$

Classify each of the equations for the following problems by degree. If the term linear, quadratic, or cubic applies, state it.

$4x+7=0$

$3y-15=9$

linear

$y=5s+6$

$y={x}^{2}+2$

$4y=8x+24$

$9z=12x-18$

linear

${y}^{2}+3=2y-6$

$y-5+{y}^{3}=3{y}^{2}+2$

cubic

${x}^{2}+x-4=7{x}^{2}-2x+9$

$2y+5x-3+4xy=5xy+2y$

$3x-7y=9$

$8a+2b=4b-8$

linear

$2{x}^{5}-8{x}^{2}+9x+4=12{x}^{4}+3{x}^{3}+4{x}^{2}+1$

$x-y=0$

linear

${x}^{2}-25=0$

${x}^{3}-64=0$

cubic

${x}^{12}-{y}^{12}=0$

$x+3{x}^{5}=x+2{x}^{5}$

fifth degree

$3{x}^{2}{y}^{4}+2x-8y=14$

$10{a}^{2}{b}^{3}{c}^{6}{d}^{0}{e}^{4}+27{a}^{3}{b}^{2}{b}^{4}{b}^{3}{b}^{2}{c}^{5}=1,\text{\hspace{0.17em}}d\ne 0$

19th degree

The expression $\frac{4{x}^{3}}{9x-7}$ is not a polynomial because

The expression $\frac{{a}^{4}}{7-a}$ is not a polynomial because

. . . there is a variable in the denominator

Is every algebraic expression a polynomial expression? If not, give an example of an algebraic expression that is not a polynomial expression.

Is every polynomial expression an algebraic expression? If not, give an example of a polynomial expression that is not an algebraic expression.

yes

How do we find the degree of a term that contains more than one variable?

Exercises for review

( [link] ) Use algebraic notation to write “eleven minus three times a number is five.”

$11-3x=5$

( [link] ) Simplify ${\left({x}^{4}{y}^{2}{z}^{3}\right)}^{5}$ .

( [link] ) Find the value of $z$ if $z=\frac{x-u}{s}$ and $x=55,\text{\hspace{0.17em}}u=49,$ and $s=3$ .

$z=2$

( [link] ) List, if any should appear, the common factors in the expression $3{x}^{4}+6{x}^{3}-18{x}^{2}$ .

( [link] ) State (by writing it) the relationship being expressed by the equation $y=3x+5$ .

$\text{The}\text{\hspace{0.17em}}\text{value}\text{​}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\text{is\hspace{0.17em}5\hspace{0.17em}more\hspace{0.17em}then\hspace{0.17em}three\hspace{0.17em}times\hspace{0.17em}the\hspace{0.17em}value\hspace{0.17em}of\hspace{0.17em}}x\text{.}$

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