Exponents and Roots in Algebra

• Algebra involves unknown values and variables represented by exponents and roots.
• Exponents can be any positive integer, and the pattern formed is X to the nth power (X^nth).
• X squared (X²) means multiplying two x's together (X x X = X²).
• The exponent and the root operation cancel each other out because they are the opposites of each other.
  E.g., √X is the opposite of X², ³√X is the opposite of X³ and so on.
• Therefore, To undo a root, use its corresponding exponent, and to undo an exponent, use its corresponding root.
  E.g., Undoing an exponent:
  
  Exponent 2:
  X²
  √X²
  = X.

  Exponent 3:
  X^{3
  3}√X³
  = X
  

  Exponent 4 :
  X^{4
  4}√X⁴
  = X
  

  And so on...
  Undoing roots:
  
  squareroot:
  √X
  (√X)²
  = X.

  cubic root:
  ³√X
  (³√X)³
  = X
  

  4th root :
  ⁴√X
  (⁴√X)⁴
  = X
  

  And so on...

Solving Equations with Roots and Exponents

• Solving algebraic equations involves getting the unknown value by itself on one side of the equal sign.
• To undo a square root, square both sides of the equation (√X = 2, (√X)² = (2)², therefore X = ±2), and to undo a cube root, cube both sides of the equation (X² = 4, √X² = √4, therefore X = ±2. ).
• Get rid of an exponent by using its inverse operation, which is a root.
• Always keep the equation in balance by doing the same operation on both sides.
• Simple algebraic equations can have more than one solution due to negative numbers.

Rules about Exponents

• Any number or variable raised to the power is 1 (X⁰= 1, 1⁰ = 1, 2⁰ = 1, etc.).
• Any number or variable raised to the power of 1 is just itself (X¹ = X, 1¹ = 1, 2¹ = 2, etc.).

Solving equations with exponents and roots

• Use a special "plus or minus sign" for even roots, since the answer could be positive or negative.
  
  X² = 4
  √X² = √4
  Therefore, X = ±2 ,
  

  X⁴ = 16
  ⁴√(X⁴) = ⁴√16
  Therefore, X = ±2

  X⁶ = 64
  ⁶√(X⁶) = ⁶√64
  Therefore, X = ±2
• To solve for odd roots, take the corresponding root of both sides of the equation.
• Negative numbers are not valid solutions for odd roots since the result is always positive.

Practice

• Rewatch the video as many times as you want and, do some exercise problems once you feel ready by clicking the 'Test Your Understanding' button.

Introduction to Polynomials

• In Algebra, terms are made up of a number part (coefficient) and a variable part.
• A polynomial is a combination of many terms linked together by addition or subtraction.
• Polynomials can be classified based on the number of terms, e.g., monomial, binomial, trinomial, and polynomial.
• The degree of a term is determined by the power of the variable part, and the degree of a polynomial is determined by the degree of its highest term.
• - Mathematicians like to arrange polynomials in order from the highest degree to the lowest.

Understanding Terms

• A term is made up of a number part (coefficient), usually an integer, and a variable part (usually a letter), which can be one or more variables with powers.
• A variable part can be raised to the power of 0, in which case it equals 1, and it's called a constant term.
  E.g., 3x + 1, 1 is the constant in this polynomial.
• The number part of a term is always written before the variable part. like this (5x) instead of (x5).
• Terms can have any number of variables, but most of the time, they have one or two variables.

The Degree of a Term

• The degree of a term is determined by the power of the variable part.
• A term with no variable part is considered a constant term with a degree of 0.
• When a term has more than one variable, the degree is the sum of the powers of each variable.
• The degree of a term helps us classify and order polynomials.

Arranging Polynomials

• Polynomials are arranged in order from the highest degree to the lowest.
• When rearranging polynomials, it's important to pay attention to the signs of the coefficients.
• All terms in a polynomial are considered added, and the operator in front of each term determines its positive or negative value.
• It's okay for a polynomial to have missing terms, and they are considered to have coefficients of 0.

Treating polynomials as combinations of positive and negative terms

• When moving a term with a negative sign to the front of the polynomial, the negative sign must come with it to maintain the term's value.
• Treating polynomials as a combination of positive and negative terms can help in re-arranging and simplifying them.

Overwhelming nature of polynomials

• It may take some time for the concepts related to polynomials to make sense.
• Re-watching the video and do practice problems can help in better understanding the topic.

Introduction

• As said on the last session, polynomials are chains of terms that are either added or subtracted together.
• Simplifying a polynomial involves identifying terms that are similar enough that they can be combined into a single term to make the polynomial shorter.

Combining Like Terms

• Terms that have the exact same variable part can be combined into a single term.
• Combining like terms involves adding the number parts and keeping the variable part the same.
  E.g., 3x + 2x -8, after adding like terms, the equation will look as follows:
  5x -8.

Identifying Like Terms

• The variable parts of a term have to be exactly the same in order to combine them.
• Like terms can be identified by looking for terms that have the same variable part.
• Examples:
  • 2x and 3x are like terms.
  • 4x and 5y are not like terms.
  • Two 'x squared' and negative seven 'x squared' are like terms.
  • Four'x squared' and six 'x cubed' are not like terms.
  • Negative 5xy and 8yx are like terms.
  • Five 'x squared y' and five 'y squared x' are not like terms.

Examples of Simplifying Polynomials

• 'x squared' plus six 'x' minus 'x' plus ten can be simplified to 'x squared' plus five 'x' plus 10.
• Sixteen minus two 'x cubed' plus four 'x' minus ten can be simplified to negative 2 'x cubed' plus four 'x' plus 6.
• Three 'x squared' plus ten minus three 'x' plus five 'x squared' minus four plus'x' can be simplified to eight 'x squared' minus two 'x' plus 6.

How to simplify polynomials

• Combine like terms by adding or subtracting coefficients.** Identify terms with the same variable(s) and exponent(s).** Cross off combined terms in the original polynomial.
• Terms that can't be combined should be written as is in the simplified polynomial.
• Treat each term as either positive or negative depending on the sign in front of it.

Importance of practicing

• It's important to practice simplifying polynomials on your own to fully understand the process.
• Re-watching the video and do practice problems can help in better understanding the topic.

Solving x by factoring

• 9x² - 12x -5
• First, find the product of a and c (A * C).
  In the above equation, a = 9 and c = -5, therefore 9 * -5 = -45

Find the factors of the product

• 45 x -1, -45 x 1, 9 x -5, -9 x 5, 15 x -3 and -15 x 3.
• Replace B, which is -12x in the above equation with factors that give you B when adding them together.
  -15 + 3 = -12, therefore 9x² - 15x + 3x -5

Grouping

• Surround the first two terms with their own paranthesis (), and do the same the last two terms.
  (9x² - 15x), (+ 3x -5)

Factorize

• Factorize both terms. When factorizing both terms, you should end up with same terms inside paranthesis. 3x(3x -5), +1(+3x -5)
• Because terms inside paranthesis are the same, take one of them and solve for x:
  3x - 5 = 0
  3x = 5
  x = 5/3.
  Then take the ones outside of the paranthesis, group them together and solve for x:
  3x + 1
  3x = -1
  x = -1/3.
  

• x² - 10x -18 = 0
• First, add 18 to both sdes. x² - 10x -18 + 18 = 0 + 18, which gives you x² - 10x = 18.
• Then, B = (b/2)², (-10/2)² = 25.
• Add this 25 on both sides of the equation, x² - 10x + 25 = 18 + 25
• x² - 10x + 25 = 45
• x² - 10x + 25 = (x + b/2)²
  = (x + (-10/2))²
  = (x - 5)²
• Therefore, (x - 5)² = 43
  
• √(x - 5)² = √43
• x - 5= √43
• x = 5 ±√43

Introduction

• The video explains the quadratic formula and its importance in Algebra 1.
• The quadratic formula is critical to know.
• The highest power of a polynomial equation determines the number of solutions it has.
• Polynomials have special characteristics, and the quadratic formula applies only to quadratic equations.

Solving Quadratic Equations

• Quadratic equations are second-degree polynomial equations.
• The first approach to solve a quadratic equation is to factor it.
• The quadratic formula is used when a polynomial equation cannot be factored.
• The quadratic formula is written in the standard form of a polynomial equation (ax^2 + bx + c = 0).

Standard Form of a Polynomial Equation

• The standard form of a polynomial equation is when the equation is written from the highest power to the lowest power.
• The coefficients of each term are assigned specific variables; a for the coefficient of x^2, b for the coefficient of x, and c for the constant value.
• The quadratic formula is applied using these values.

Applying the Quadratic Formula

• The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
• The values of a, b, and c are substituted into the quadratic formula.
• The signs of negative values are put in parentheses to avoid confusion.
• The square root part of the formula goes all the way over the B^2-4ac part.
• The quadratic formula is then solved using the order of operations.
• Double-checking the values that were plugged into the formula is essential to avoid mistakes.

Steps to Solve a Quadratic Equation

• Assign values to the variables a, b, and c.
• Use the quadratic formula:
  x = (-b ± √(b^2 - 4ac)) / 2a
• Plug in values correctly using parentheses.
• Simplify step by step, being mindful of the B² - 4AC part which is a common place where students make mistakes.
• The plus or minus part of the formula gives the two solutions
• Calculate each solution separately

Common Mistakes

• Students often confuse signs and make errors with the B² - 4AC part of the formula.
• Using parentheses correctly is crucial to avoid mistakes.
• Re-watching the video and do practice problems can help in better understanding the topic.