Brief Summary
This video provides a comprehensive review of essential algebra concepts, including combining like terms, polynomial operations, exponent properties, solving equations, and graphing linear equations. It offers numerous examples and practice problems to reinforce understanding.
- Combining like terms involves adding coefficients of terms with the same variable and exponent.
- Polynomial operations include addition, subtraction, and multiplication, with special attention to distributing monomials and binomials.
- Exponent properties cover multiplication, division, and raising powers to powers, including negative exponents.
- Solving equations involves isolating variables using inverse operations and techniques like factoring and the quadratic formula.
- Graphing linear equations is explained using slope-intercept and standard forms, including finding equations of parallel and perpendicular lines.
Combining Like Terms
Like terms can be combined by adding their coefficients. For example, 5x + 4x = 9x. In expressions with multiple terms, only like terms can be combined. For instance, in 3x + 4y + 5x + 8y, combine 3x and 5x to get 8x, and 4y and 8y to get 12y, resulting in 8x + 12y. Similarly, terms with the same radical can be combined, such as 3√2 + 8√2 = 11√2 and 5√7 + 3√7 = 8√7.
Adding and Subtracting Polynomials
A polynomial is an expression with multiple terms; a monomial has one term (e.g., 8x), a binomial has two terms (e.g., 5x + 6), and a trinomial has three terms (e.g., x^2 + 6x + 5). When subtracting polynomials, distribute the negative sign to all terms in the second polynomial before combining like terms. For example, to simplify 3x^2 + 7x - 4 - (8x^2 - 5x + 7), distribute the negative sign to get 3x^2 + 7x - 4 - 8x^2 + 5x - 7, then combine like terms to get -5x^2 + 12x - 11.
Multiplying Polynomials
To multiply a monomial by a trinomial, distribute the monomial to each term inside the trinomial. For example, 7x * (x^2 + 2x - 3) involves multiplying 7x by x^2, 2x, and -3, resulting in 7x^3 + 14x^2 - 21x. When multiplying a binomial by another binomial, use the FOIL method (First, Outer, Inner, Last) to ensure each term is multiplied correctly. For instance, (3x - 4) * (2x + 7) becomes 6x^2 + 21x - 8x - 28, which simplifies to 6x^2 + 13x - 28.
Special Case: Squaring a Binomial
To simplify (2x - 3)^2, rewrite it as (2x - 3) * (2x - 3) and then use the FOIL method. This gives 4x^2 - 6x - 6x + 9, which simplifies to 4x^2 - 12x + 9.
Multiplying Binomials and Trinomials
When multiplying a binomial by a trinomial, each term in the binomial must be multiplied by each term in the trinomial, resulting in six terms before simplification. For example, (5x - 9) * (2x^2 - 3x + 4) expands to 10x^3 - 15x^2 + 20x - 18x^2 + 27x - 36. Combining like terms gives the final answer: 10x^3 - 33x^2 + 47x - 36.
Properties of Exponents
When multiplying variables with exponents, add the exponents: x^3 * x^4 = x^7. When dividing, subtract the exponents: x^9 / x^4 = x^5. When raising an exponent to another exponent, multiply them: (x^7)^6 = x^42. A negative exponent indicates a reciprocal: x^-3 = 1/x^3.
Exponent Rules with Coefficients
When multiplying terms with coefficients and exponents, multiply the coefficients and add the exponents of like variables. For example, (3x^4y^5) * (5x^6y^7) = 15x^10y^12. When dividing, divide the coefficients and subtract the exponents. For instance, (24x^7y^-2) / (6x^4y^5) simplifies to 4x^3y^-7, which is then rewritten as 4x^3 / y^7 to eliminate the negative exponent.
Raising Expressions to a Power
When raising an expression with coefficients and exponents to a power, distribute the power to each factor. For example, (3x^3)^2 = 3^2 * x^6 = 9x^6. Anything raised to the zero power is one, except when a negative coefficient is outside the parentheses, such as -2 * (5xy^3)^0 = -2 * 1 = -2.
Simplifying Complex Fractions with Exponents
To simplify complex fractions with negative exponents, first move terms with negative exponents to the opposite side of the fraction to make the exponents positive. Then, simplify by multiplying or dividing coefficients and adding or subtracting exponents. For example, to simplify (5x^-2 / y^-3) * (8x^4 / y^5), rewrite it as (5y^3 / x^2) * (8x^4 / y^5), which simplifies to 40x^2 / y^2.
Simplifying Fractions with Multiple Variables
Simplify fractions by breaking down coefficients into smaller factors and canceling common terms. For example, simplify (35x^-3 / 40xy^5) * (24x^2y^2 / 42y^4) by breaking down the numbers and canceling common factors, resulting in (y / (2x^2)). When dividing fractions, use "Keep, Change, Flip" to multiply by the reciprocal of the second fraction.
Solving Linear Equations
To solve for x, isolate the variable by performing inverse operations on both sides of the equation. For example, to solve x + 4 = 9, subtract 4 from both sides to get x = 5. For more complex equations like 3x + 5 = 11, first subtract 5 from both sides to get 3x = 6, then divide by 3 to find x = 2.
Solving Multi-Step Equations
To solve multi-step equations, simplify by distributing and combining like terms before isolating the variable. For example, in 2(x - 1) + 6 = 10, subtract 6 from both sides to get 2(x - 1) = 4, then divide by 2 to get x - 1 = 2, and finally add 1 to find x = 3. In equations with variables on both sides, such as 5 - 3(x + 4) = 7 + 2(x - 1), distribute and combine like terms, then move all x terms to one side and constants to the other before solving for x.
Solving Equations with Fractions
To solve equations with fractions, eliminate the fractions by multiplying both sides by the common denominator. For example, to solve (3/4)x - (1/3) = 1, multiply both sides by 12 to get 9x - 4 = 12, then solve for x. When given two fractions separated by an equal sign, cross multiply to solve for the variable.
Solving Equations with Decimals
To solve equations with decimals, eliminate the decimals by multiplying both sides by a power of 10. Choose the power of 10 based on the maximum number of digits after the decimal point in the equation. For example, if the equation has two digits after the decimal, multiply by 100.
Solving Quadratic Equations by Factoring
To solve quadratic equations, rearrange the equation to equal zero, then factor the quadratic expression. Set each factor equal to zero and solve for x. For example, to solve x^2 - 5x + 6 = 0, factor it into (x - 2)(x - 3) = 0, then set x - 2 = 0 and x - 3 = 0 to find x = 2 and x = 3.
Factoring Quadratics with Leading Coefficient Not Equal to One
When the leading coefficient is not one, multiply the leading coefficient by the constant term, find two numbers that multiply to this product and add to the middle term, replace the middle term with these two numbers, and factor by grouping.
Using the Quadratic Formula
When factoring is difficult, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). Identify a, b, and c from the quadratic equation in standard form (ax^2 + bx + c = 0), plug these values into the formula, and simplify to find the solutions for x.
Solving Cubic Functions by Factoring
To solve cubic functions, try factoring by grouping. Look for common factors in pairs of terms and factor them out. If a common binomial factor emerges, factor it out, and then solve for x by setting each factor equal to zero.
Graphing Linear Equations in Slope-Intercept Form
To graph a linear equation in slope-intercept form (y = mx + b), identify the slope (m) and y-intercept (b). Plot the y-intercept on the y-axis, then use the slope (rise over run) to find additional points. Connect the points to draw the line.
Graphing Linear Equations in Standard Form
To graph a linear equation in standard form (Ax + By = C), find the x and y intercepts. Substitute 0 for y to find the x-intercept and 0 for x to find the y-intercept. Plot these points and connect them with a straight line.
Writing Equations of Lines
To write the equation of a line given a point and a slope, use the point-slope form (y - y1 = m(x - x1)). Substitute the given point (x1, y1) and slope (m) into the equation. Convert to slope-intercept form (y = mx + b) by distributing and solving for y. Convert to standard form (Ax + By = C) by rearranging the equation.
Finding Equations with Two Points
To write the equation of a line given two points, first find the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Then, use the point-slope form with one of the points and the calculated slope to write the equation. Convert to slope-intercept form and then to standard form as needed.
Parallel and Perpendicular Lines
To find the equation of a line parallel to a given line, determine the slope of the given line by converting it to slope-intercept form. Parallel lines have the same slope. Use this slope and a given point to write the equation of the parallel line. For a perpendicular line, use the negative reciprocal of the slope of the given line.
