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Algebra Introduction - Basic Overview - Online Crash Course Review Video Tutorial Lessons

The Organic Chemistry Tutor・53 minutes read

Algebraic operations involve combining like terms, simplifying expressions, and understanding exponent rules, with various examples provided for solving equations and factoring polynomials. Graphing linear equations requires knowledge of slope-intercept form and the ability to plot points to create a line.

Insights

  • Combining like terms in algebra involves adding coefficients of similar variables, simplifying expressions like 5x + 4x = 9x.
  • Polynomials can be simplified by subtracting or adding coefficients of like terms, distributing negative signs when subtracting, and understanding the number of terms in monomials, binomials, and trinomials.
  • Exponent rules are crucial in algebra, requiring operations like multiplying exponents when raising to a power, subtracting exponents when dividing, and understanding that any expression raised to the zero power equals one.

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Recent questions

  • What are like terms in algebra?

    Like terms share the same variables.

  • How can polynomials be simplified?

    Combine like terms and distribute negative signs.

  • What is the FOIL method in algebra?

    FOIL stands for First, Outer, Inner, Last.

  • How are exponents handled in algebraic expressions?

    Exponents are added, subtracted, or multiplied.

  • How can algebraic equations be solved?

    Use factoring, distribution, and simplification.

Related videos

Summary

00:00

Algebraic Operations: Combining Like Terms and More

  • Like terms in algebra allow for the addition of coefficients, such as 5x + 4x resulting in 9x.
  • Unlike terms cannot be added, but like terms can be combined, as seen in 3x + 5x = 8x.
  • Radical terms can also be combined if they share the same radical, like 3√2 + 8√2 = 11√2.
  • Adding like terms with different exponents, such as 4x² + 9x² = 13x², is a common algebraic operation.
  • When subtracting polynomials, distribute the negative sign and combine like terms to simplify the expression.
  • Polynomials consist of multiple terms, with monomials having one term, binomials two, and trinomials three.
  • Multiplying a monomial by a trinomial involves distributing the monomial to each term in the trinomial.
  • The FOIL method is used to multiply two binomials, combining like terms to obtain the final result.
  • Simplifying expressions like (2x - 3)² requires applying the FOIL method and combining like terms.
  • Understanding exponent properties is crucial, including rules for multiplication, division, and raising exponents to other exponents.

20:33

Simplify Exponential Expressions with Division and Multiplication

  • To simplify the expression y to the negative three z to the fourth divided by 40 x to the negative 8 y to the negative 7 z to the negative 8, divide 32 by 40 by dividing both numbers by 8, resulting in 4 and 5 respectively.
  • For x to the fifth power divided by x to the negative 8, subtract the exponents (5 - (-8) = 13), placing x to the 13th power on top.
  • When dividing y to the negative three by y to the negative seven, subtract the exponents (negative 3 - negative 7 = 4), leading to y to the fourth power.
  • For z's, subtract the exponents (4 - negative 8 = 12), resulting in z to the 12th power.
  • When raising three x cubed to the second power, multiply the exponents (3 * 2 = 6), yielding nine x to the sixth.
  • Differentiate between 5x squared and 5 plus x squared, as 5x squared equals 25 x squared, while 5 plus x squared requires expanding through FOIL.
  • To find four x squared y to the third raised to the third power, calculate 4 to the third power as 64, resulting in 64 x to the sixth y to the ninth.
  • Any expression raised to the zero power equals one, as seen with eight x squared y to the fifth z to the sixth raised to the zero power.
  • Simplify five x to the negative two divided by y to the negative three times eight x to the four divided by y to the negative five by converting negative exponents and performing the necessary operations, resulting in forty x squared y to the eighth.
  • For 35 x to the negative 3 divided by 40 x y to the fifth times 24 x squared y squared divided by 42 y to the negative four, simplify by breaking down numbers, canceling out common factors, and performing the operations, leading to the final answer of 10 x squared y to the eighth over 9.

41:09

Algebraic Equations: Solving for Unknowns

  • 36 divided by 4 equals 9, which is also the result of 12 divided by 4, multiplied by 3.
  • The value of x in the equation 12x = 1/3 is 16/9 after solving for x.
  • To solve x + 2/5 = 7/8, cross multiply to get 5 * 7 = 35 and 8 * (x - 2) = 8x + 16, leading to x = 19/8.
  • When dealing with decimal numbers, multiply both sides by 100 to simplify calculations.
  • In the equation 0.04x + 0.15 - 0.09x + 0.25 = 0, simplify to 5x = 40, resulting in x = 8.
  • Solving x^2 = 25 gives x = ±5, using the square root method or factoring as (x + 5)(x - 5) = 0.
  • For 2x^2 - 18 = 0, factor out the greatest common factor (GCF) of 2 to get x = ±3.
  • In x^2 - 9 = 0, factor as (x + 3)(x - 3) = 0, leading to x = ±3.
  • To solve 3x^2 - 48 = 0, factor out the GCF of 3 to get x = ±4.
  • For x^4 - 81 = 0, factor as (x^2 + 9)(x^2 - 9) = 0, resulting in x = ±3i or ±3.

01:00:11

Mathematical Operations and Graphing Linear Equations

  • 24 multiplied by 3 equals 72, with two negative signs resulting in a positive 72.
  • 2 multiplied by 6 equals 12.
  • The sum of 49 and 72 is calculated manually as 121.
  • The square root of 121 is 11, leading to -7 plus or minus 11 divided by 12.
  • This expression can be separated into two fractions: -7 plus 11 over 12 and -7 minus 11 over 12, resulting in 4 and -18 respectively.
  • Simplifying, 4 over 12 reduces to 1 over 3.
  • Similarly, dividing 18 by 6 gives 3 and 12 by 6 gives 2, leading to the answers of -3 over 2 and 1/3.
  • The cubic function x cubed minus 4x squared minus x plus 4 equals zero can be solved by factoring.
  • Graphing linear equations involves understanding slope-intercept form, with examples provided for graphing equations like y equals 2x minus 1 and y equals 3/4x minus 2.
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