### Key Concepts

### 5.1 Add and Subtract Polynomials

**Monomial**- A
**monomial**is an algebraic expression with one term. - A monomial in one variable is a term of the form $a{x}^{m},$ where
*a*is a constant and*m*is a whole number.

- A
**Polynomials****Polynomial**â€”A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.**monomial**â€”A polynomial with exactly one term is called a monomial.**binomial**â€” A polynomial with exactly two terms is called a binomial.**trinomial**â€”A polynomial with exactly three terms is called a trinomial.

**Degree of a Polynomial**- The
**degree of a term**is the sum of the exponents of its variables. - The
**degree of a constant**is 0. - The
**degree of a polynomial**is the highest degree of all its terms.

- The

### 5.2 Properties of Exponents and Scientific Notation

**Exponential Notation**

This is read*a*to the ${m}^{th}$ power.

In the expression ${a}^{m}$, the*exponent m*tells us how many times we use the*base a*as a factor.**Product Property for Exponents**

If*a*is a real number and*m*and*n*are integers, then

$${a}^{m}\xc2\xb7{a}^{n}={a}^{m+n}$$

To multiply with like bases, add the exponents.**Quotient Property for Exponents**

If $a$ is a real number, $a\xe2\u20300,$ and*m*and*n*are integers, then

$$\frac{{a}^{m}}{{a}^{n}}={a}^{m\xe2\u02c6\u2019n},\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}m>n\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n\xe2\u02c6\u2019m}},\phantom{\rule{0.4em}{0ex}}n>m$$**Zero Exponent**- If
*a*is a non-zero number, then ${a}^{0}=1.$ - If
*a*is a non-zero number, then*a*to the power of zero equals 1. - Any non-zero number raised to the zero power is 1.

- If
**Negative Exponent**- If
*n*is an integer and $a\xe2\u20300,$ then ${a}^{\text{\xe2\u02c6\u2019}n}=\frac{1}{{a}^{n}}$ or $\frac{1}{{a}^{\text{\xe2\u02c6\u2019}n}}={a}^{n}.$

- If
**Quotient to a Negative Exponent Property**

If $a,b$ are real numbers, $a\xe2\u20300,b\xe2\u20300$ and $n$ is an integer, then

$${\left(\frac{a}{b}\right)}^{\text{\xe2\u02c6\u2019}n}={\left(\frac{b}{a}\right)}^{n}$$**Power Property for Exponents**

If $a$ is a real number and $m,n$ are integers, then

$${({a}^{m})}^{n}={a}^{m\xc2\xb7n}$$

To raise a power to a power, multiply the exponents.**Product to a Power Property for Exponents**

If*a*and*b*are real numbers and*m*is a whole number, then

$${(ab)}^{m}={a}^{m}{b}^{m}$$

To raise a product to a power, raise each factor to that power.**Quotient to a Power Property for Exponents**

If $a$ and are real numbers, $b\xe2\u20300,$ and $m$ is an integer, then

$${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$$

To raise a fraction to a power, raise the numerator and denominator to that power.**Summary of Exponent Properties**

If*a*and*b*are real numbers, and*m*and*n*are integers, then

Property Description Product Property ${a}^{m}\xc2\xb7{a}^{n}={a}^{m+n}$ Power Property ${\left({a}^{m}\right)}^{n}={a}^{m\xc2\xb7n}$ Product to a Power ${\left(ab\right)}^{n}={a}^{m}{b}^{m}$ Quotient Property $\frac{{a}^{m}}{{a}^{n}}={a}^{m\xe2\u02c6\u2019n},a\xe2\u20300$ Zero Exponent Property ${a}^{0}=1,a\xe2\u20300$ Quotient to a Power Property: ${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},b\xe2\u20300$ Properties of Negative Exponents ${a}^{\text{\xe2\u02c6\u2019}n}=\frac{1}{{a}^{n}}$ and $\frac{1}{{a}^{\text{\xe2\u02c6\u2019}n}}={a}^{n}$ Quotient to a Negative Exponent ${\left(\frac{a}{b}\right)}^{\text{\xe2\u02c6\u2019}n}={\left(\frac{b}{a}\right)}^{n}$ **Scientific Notation**

A number is expressed in scientific notation when it is of the form

$$a\phantom{\rule{0.2em}{0ex}}\xc3\u2014\phantom{\rule{0.2em}{0ex}}{10}^{n}\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}1\xe2\u2030\xa4a<10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{is an integer.}$$**How to convert a decimal to scientific notation.**- Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
- Step 2. Count the number of decimal places, $n,$ that the decimal point was moved.
- Step 3. Write the number as a product with a power of 10. If the original number is.
- greater than 1, the power of 10 will be ${10}^{n}.$
- between 0 and 1, the power of 10 will be ${10}^{\text{\xe2\u02c6\u2019}n}.$

- Step 4. Check.

**How to convert scientific notation to decimal form.**- Step 1. Determine the exponent, $n,$ on the factor 10.
- Step 2. Move the decimal $n$ places, adding zeros if needed.
- If the exponent is positive, move the decimal point $n$ places to the right.
- If the exponent is negative, move the decimal point $\left|n\right|$ places to the left.

- Step 3. Check.

### 5.3 Multiply Polynomials

**How to use the FOIL method to multiply two binomials.**

**Multiplying Two Binomials:**To multiply binomials, use the:- Distributive Property
- FOIL Method

**Multiplying a Polynomial by a Polynomial:**To multiply a trinomial by a binomial, use the:- Distributive Property
- Vertical Method

**Binomial Squares Pattern**

If*a*and*b*are real numbers,**Product of Conjugates Pattern**

If $a,b$ are real numbers

The product is called a difference of squares.

To multiply conjugates, square the first term, square the last term, write it as a difference of squares.**Comparing the Special Product Patterns**

Binomial Squares Product of Conjugates ${\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}$ ${\left(a\xe2\u02c6\u2019b\right)}^{2}={a}^{2}\xe2\u02c6\u20192ab+{b}^{2}$ $\left(a\xe2\u02c6\u2019b\right)\left(a+b\right)={a}^{2}\xe2\u02c6\u2019{b}^{2}$ â€¢â€ƒâ€ƒSquaring a binomial â€¢â€ƒâ€ƒMultiplying conjugates â€¢â€ƒâ€ƒProduct is a **trinomial**â€¢â€ƒâ€ƒProduct is a **binomial.**â€¢â€ƒâ€ƒInner and outer terms with FOIL are **the same.**â€¢â€ƒâ€ƒInner and outer terms with FOIL are **opposites.**â€¢â€ƒâ€ƒMiddle term is **double the product**of the termsâ€¢â€ƒâ€ƒThere is **no**middle term.**Multiplication of Polynomial Functions:**- For functions $f\left(x\right)$ and $g(x),$

$$\left(f\xc2\xb7g\right)\left(x\right)=f\left(x\right)\xc2\xb7g\left(x\right)$$

- For functions $f\left(x\right)$ and $g(x),$

### 5.4 Dividing Polynomials

**Division of a Polynomial by a Monomial**- To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

**Division of Polynomial Functions**- For functions $f\left(x\right)$ and $g(x),$ where $g(x)\xe2\u20300,$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

- For functions $f\left(x\right)$ and $g(x),$ where $g(x)\xe2\u20300,$
**Remainder Theorem**- If the polynomial function $f\left(x\right)$ is divided by $x\xe2\u02c6\u2019c,$ then the remainder is $f\left(c\right).$

**Factor Theorem:**For any polynomial function $f\left(x\right),$- if $x\xe2\u02c6\u2019c$ is a factor of $f\left(x\right),$ then $f\left(c\right)=0$
- if $f\left(c\right)=0,$ then $x\xe2\u02c6\u2019c$ is a factor of $f\left(x\right)$