4.9: Summary of Key Concepts
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Algebraic Expressions
An algebraic expression (often called simply an expression) is a number, a letter, or a collection of numbers and letters along with meaningful signs of operation. (5÷0 is not meaningful.)
Terms
In an algebraic expression, the quantities joined by "+" signs are terms.
Distinction Between Terms and Factors
Terms are parts of sums and are therefore separated by addition signs. Factors are parts of products and are therefore separated by multiplication signs.
Common Factors
In an algebraic expression, a factor that appears in every term, that is, a factor that is common to each term, is called a common factor.
Coefficients
The coefficient of a quantity records how many of that quantity there are. The coefficient of a group of factors is the remaining group of factors.
Distinction Between Coefficients and Exponents
Coefficients record the number of like terms in an expression.
\(\underbrace{x+x+x}_{3 \text { terms }}=\begin{array}{c}
3 x \\
\text { coefficient is } 3
\end{array}\)
Exponents record the number of like factors in an expression
\(\underbrace{x \cdot x \cdot x}_{3 \text { factors }}=\begin{array}{c}
x^{3} \\
\text { exponent is } 3
\end{array}\)
Equation
An equation is a statement that two expressions are equal.
Numerical Evaluation
Numerical evaluation is the process of determining a value by substituting numbers for letters.
Polynomials
A polynomial is an algebraic expression that does not contain variables in the denominators of fractions and in which all exponents on variable quantities are whole numbers.
A monomial is a polynomial consisting of only one term.
A binomial is a polynomial consisting of two terms.
A trinomial is a polynomial consisting of three terms.
Degree of a Polynomial
The degree of a term containing one variable is the value of the exponent on the variable.
The degree of a term containing more than one variable is the sum of the exponents on the variables.
The degree of a polynomial is the degree of the term of the highest degree.
Linear Quadratic Cubic Polynomials
Polynomials of the first degree are linear polynomials.
Polynomials of the second degree are quadratic polynomials.
Polynomials of the third degree are cubic polynomials.
Like Terms
Like terms are terms in which the variable parts, including the exponents, are identical.
Descending Order
By convention, and when possible, the terms of an expression are placed in descending order with the highest degree term appearing first. \(5x^3−2x^2+10x−15\) is in descending order.
Multiplying a Polynomial by a Monomial
To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.
\(7(x−3)=7x−7 \cdot 3=7x−21\)
Simplifying \(+(a+b)\) and \(−(a+b)\)
\(\begin{array}{l}
+(a+b)=a+b \\
-(a+b)=-a-b
\end{array}\)
Multiplying a Polynomial by a Polynomial
To multiply polynomials together, multiply every term of one polynomial by every term of the other polynomial.
\(\begin{aligned}
(x+3)(x-4) &=x^{2}-4 x+3 x-12 \\
&=x^{2}-x-12
\end{aligned}\)
Special Products
\(\begin{aligned}
(a+b)^{2} &=a^{2}+2 a b+b^{2} & \text { Note }: &(a+b)^{2} \neq a^{2}+b^{2} \\
(a-b)^{2} &=a^{2}-2 a b+b^{2} &(a-b)^{2} \neq a^{2}-b^{2} \\
(a+b)(a-b) &=a^{2}-b^{2} & &
\end{aligned}\)
Independent and Dependent Variables
In an equation, any variable whose value can be freely assigned is said to be an independent variable. Any variable whose value is determined once the other values have been assigned is said to be a dependent variable.
Domain
The collection of numbers that can be used as replacements for the independent variable in an expression or equation and yield a meaningful result is called the domain of the expression or equation.