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Evaluating Expressions
Algebraic Expression: An expression composed of variables and/or numerals. Algebraic expressions may also include grouping symbols and/or operation signs.
Variable: A letter that represents an unknown number.
Constant: A number that never changes.
Evaluate: To replace each variable with a number and then calculate the result.
Value: The calculated number that is the result of evaluating an expression.
Coefficient: A number that multiplies a variable.
If the coefficient is -1, it can be represented by a negative sign directly in front of the variable.
Terms: The components that make up algebraic expressions. Terms include numbers, variables, products of numbers and/or variables, and quotients of numbers and/or variables.
Like Terms: Terms that contain the same variable(s) raised to the same power(s).
To simplify an expression, it is necessary to combine like terms.
Simplifying: Writing a simpler, equivalent expression by combining like terms, canceling common factors, combining and simplifying fractions, and applying the Distributive, Associative, and Commutative Laws.
The Commutative and Associative Laws
The Commutative and Associative Laws are used to simplify expressions.
Solving Equations
Equation: A number sentence with an equal sign between two expressions.
Solution: Any replacement for a variable that makes an equation or inequality true.
When canceling a coefficient that is a fraction, it can either be divided by itself or multiplied by its reciprocal.
Like terms should be combined before solving an equation. If like terms are on the same side, combine them. If like terms are on both sides, move one over, then combine.
Every equation can be classified as an identity, a contradiction, or a conditional equation.
Identity: An equation that is true for all replacements.
Contradiction: An equation that is never true.
Conditional: An equation that is true for some replacements.
Solution Set: The set of all solutions.
Formula: An equation that uses numbers and/or letters to represent a relationship between two or more quantities.
Solving Inequalities
Inequality: A mathematical sentence using ≤ , ≥ , < , > , or ≠.
The same steps used to solve equations are used to solve inequalities.
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When multiplying or dividing both sides of an inequality by a negative value, the direction of the inequality sign is flipped.
The > and < inequality symbols are graphed with an open circle to indicate that the endpoint is not a solution. The ≥ and ≤ symbols are graphed with a closed circle to indicate that the endpoint is a solution.
A variable can also be located between two inequality symbols.
Set-Builder Notation: A set is named by describing the characteristics of its elements.
Interval Notation: A set of all numbers is represented by a pair of numbers inside parentheses (> and <) and/or brackets (≥ and ≤). When graphing inequalities, brackets and parentheses can be used instead of circles.
Word Problems
When working with word problems, certain key words can indicate a specific operation, inequality, or equality.
Coordinate Plane
Coordinate Plane: A plane used to graph points, lines, and curves. A coordinate plane is composed of a horizontal number line, the first axis, and a vertical number line, the second axis.
Axes: Two perpendicular number lines that are used to identify points in a plane.
Origin: The point (0 , 0) on a coordinate plane. The origin is the location where the two axes intersect.
Ordered Pair: A pair of numbers in the form (x , y).
Coordinates: The numbers in an ordered pair.
Abscissa: The first coordinate that determines the horizontal distance from the origin.
Ordinate: The second coordinate that determines the vertical distance from the origin.
The variables x and y are typically used when graphing on a plane.
Quadrants: A coordinate plane has four regions. Coordinates in each region carry different signs.
A coordinate plane can be used to visualize solutions of a given equation. Every solution is represented by a point on the coordinate plane.
Linear Equation: In two variables, an equation whose graph is a straight line. Linear equations are commonly written in the form of y=mx+b or Ax+By=C.
To determine if an equation is linear, three solutions must be found, and a single straight line must pass through these solutions.
y-axis: The vertical axis in a coordinate plane.
x-axis: The horizontal axis in a coordinate plane.
y-intercept: The point at which a graph crosses the y-axis
x-intercept: The point at which a graph crosses the x-axis
Slope: The ratio of the vertical change to horizontal change that measures a line’s slant.
It does not matter which ordered pair is considered (x1, y1) or (x2 , y2); the simplified fraction will be the same regardless.
A line with a positive slope slants up from left to right, and a line with a negative slope slants down from left to right. Also, the larger the absolute value of the slope, the steeper the line.
Horizontal Line: A line with a slope of zero.
Vertical Line: A line with a slope that is undefined.
Excluding vertical and horizontal lines, there are three common forms of linear equations.
Two different lines are parallel if they have the same slope or if both of the lines are vertical.
Two lines are perpendicular if the product of their slopes is -1 or if one line is horizontal and the other line is vertical. The perpendicular line’s slope can be determined by calculating the negative reciprocal of the given line’s slope.
Rate: A measure of how two quantities change with respect to each other.
Rates can be visualized using a coordinate plane.
Interpolation: Estimating a value between given values.
Extrapolation: Estimating a value beyond the given data.
Polynomials
Monomial: The product of constants and/or variables. Monomials do not include division by a variable.
Polynomial: A monomial or a sum of monomials.
Terms of the Polynomial: When a polynomial is written as a sum of monomials, each monomial is called a term of the polynomial.
Binomial: A polynomial with two terms.
Trinomial: A polynomial with three terms.
The Degree of a Monomial: The number of variable factors in the monomial.
Leading Term: For a polynomial, the highest degree term.
Leading Coefficient: The coefficient of the leading term.
The Degree of the Polynomial: The degree of the leading term.
Descending Order: The terms of a polynomial are arranged according to degree, from greatest to least. Descending order is the standard way polynomials are written.
Like terms within polynomials should be combined.
Polynomials can be evaluated if each variable is replaced with a number.
Polynomials are added together by using the commutative, associative, and distributive laws.
Polynomials are subtracted by adding the opposite of the polynomial being subtracted.
Monomials are multiplied by multiplying the coefficients and applying the product rule for exponents to the variables
To multiply a monomial by a polynomial, the Distributive Law must be applied.
To multiply polynomials, the Distributive Law must be applied multiple times.
The FOIL method can be used to multiply binomials.
The product of the sum and the difference of the same two terms can be simplified.
The square of a binomial can be simplified.
A polynomial can be divided by a monomial by using the quotient rule.
For divisors with more than one term, long division is required.
After performing an operation involving polynomials and/or monomials, check your result by substituting values for the variables in both the original expression and new expression. If the outputs are equal, the operation was completed correctly.
Factoring Polynomials
Factoring: To find an equal expression that is a product.
To factor a monomial, we find monomials whose product is the same as the original monomial.
Largest Common Factor of a Polynomial: The largest common factor of the coefficients multiplied by the largest common factor of the variables in all terms. Use prime factorization to find the largest common factor. Unlike the greatest common factor, the largest common factor may represent a negative.
If terms in a polynomial have a common factor, the largest common factor is factored out.
Factoring by Grouping: A polynomial is factored by splitting terms into two groups that share a common factor. This method is used on polynomials with four or more terms. Swapping terms between groups may be necessary to determine the common factor.
Nonprime trinomials with a leading coefficient of 1 can be factored into two binomials.
Prime Polynomial: A polynomial that cannot be factored using rational numbers.
To factor some polynomials completely, more than one factoring method may be required.
Nonprime trinomials with a leading coefficient other than 1 can be factored using FOIL.
Nonprime trinomials with a leading coefficient other than 1 can be factored using grouping.
Perfect square trinomials can be factored by a unique property.
Differences of squares can be factored by a unique property. There is no general equation for factoring a sum of squares.
Differences and sums of cubes can be factored by a unique property.
Quadratic Equation: An equation in which a second-degree polynomial is equal to zero. Quadratic equations can be solved by factoring.
The solutions for a quadratic equation can be checked by graphing.
Rational Expressions
Rational Expressions: Expressions that can be written as the quotient of two polynomials.
Any replacement values that result in the denominator equaling zero must be avoided.
Rational expressions are simplified by first factoring the numerator and/or the denominator and then canceling out any common factors.
To multiply rational expressions, first combine the rational expressions so they share a single fraction bar and factor each polynomial if possible; then, cancel any like terms that appear in the numerator and denominator.
To divide rational expressions, multiply by the reciprocal.
To add rational expressions with the same denominator, first combine the rational expressions so they share a single fraction bar and add; then, factor and cancel any like terms that appear in the numerator and denominator.
To subtract rational expressions with the same denominator, first combine the rational expressions so they share a single fraction bar, convert the subtrahend to its opposite, and add; then, factor and cancel any like terms that appear in the numerator and denominator.
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To add or subtract rational expressions with different denominators, finding the least common denominator will be required. If the denominators have factors in common, include each shared factor only once.
To add rational expressions with different denominators: first, factor each polynomial if possible; second, find the least common denominator and rewrite each expression with it; third, add the numerators; finally, factor the resultant polynomial if possible and cancel any like terms that appear in the numerator and denominator.
To subtract rational expressions with different denominators: first, factor each polynomial if possible; second, find the least common denominator and rewrite each expression with it; third, convert the subtrahend to its opposite and add the numerators; finally, factor the resultant polynomial if possible and cancel any like terms that appear in the numerator and denominator.
When adding or subtracting rational expressions, if one denominator is the opposite of the other, multiply one of the rational expressions by -1/-1.
Complex Rational Expressions: Rational expressions that have one or more rational expressions within their numerator and/or denominator.
To simplify a complex rational expression, the numerator and denominator need to be added or subtracted so that they both contain a single rational expression. Then, the numerator should be multiplied by the reciprocal of the denominator.
Rational Equation: An equation that contains one or more rational expressions.
When solving a rational equation: clear fractions by multiplying both sides by the least common denominator, solve the resultant equation, and check that solutions are not restricted values by substituting them into the original equation.
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