MATH CHEAT SHEET: ALGEBRA, GEOMETRY, TRIGONOMETRY


With the basics and the more practical uses of math under their belts, high school students start to expand their general understanding of mathematics. Many math terms sound scary, but polynomials, functions and variables are fairly straightforward concepts that can be mastered through practice.

Because it’s unlikely you use polynomials everyday, you’ve probably forgotten much of what your child will be studying.

“It is OK not to be an expert in the subject that your child is studying, but you do need to be an expert on communication with your child’s teachers,” says Richard E. Bavaria, Ph.D., vice president of education for Sylvan Learning Center. “It is important to know when examinations will take place and what your child will need to prepare.”

Between sessions with your child’s teacher, here’s a refresher course to get your mind back in the game.

Algebra is a method of solving for an unknown number, or variable, which is represented by a letter. The easiest way to work through an algebra equation is to think of it as a seesaw where both sides have to balance. Whatever you do to one side to solve for x, do it to the other side.

Example: x + 3 = 10
x + 3 – 3 = 10 – 3
    x = 7
3x – 1 = 20
3x – 1 + 1 = 20 + 1
      3x = 21
      3      3
      x = 7

Functions: A mathematical relationship between two variables. Functions are notated:
 
f(x) = y
f(x) = 3y – 4
f(x) = 4y2 – 3y + 2

To solve for f(5) just plug in the number in parentheses into the y variable.
 
f(x) = y
f(5) = 5
f(x) = 3y - 4
f(5) = (3)(5) - 4
f(5) = 11
f(x) = 4y2 – 3y + 2
f(5) = (4)(52) – 3(5) + 2
f(5) = 100 – 15 + 2
f(5) = 87

Absolute Value is the distance a number is from 0.
The absolute value of -5 is 5. |-5| = 5
The absolute value of 5 is 5. |5| = 5

Factoring: A factor is a number that is multiplied by another number to create a product. For example: 3 × 3 = 9, 3 is a factor and 9 is a product. Notice that this can also be written as 32 = 9. If it were an algebraic equation: x2 = 9 then x = 3 or -3. You find this answer by taking the square root of both sides. Because -3 × -3 = 9 or (-3)2 = 9 there are two answers.

Polynomials are mathematical expressions involving the sum of variables raised to a certain power.
  • Monomials: 3x 4x2 10x2y3
  • Binomials: 2x – 1 3x2 + 6y 4(x2 + 6)
  • Trinomials: 3x2 – 3x + 6 y2 + 2y – 4 z2 – 1

A trinomial is the product of two binomials. To multiply two binomials follow this formula:
  (x – a)(x – b) x2 – ax – bx + ab
  (x – a)(x + b) x2 – ax + bx – ab
  (x + a)(x – b) x2 + ax – bx – ab
  (x + a)(x + b) x2 + ax + bx + ab

Factoring trinomials is the opposite of multiplying binomials.
x2 – 4x + 4 = (x – 2)(x – 2) = (x – 2)2

In this example, (x – 2) is a factor and x2 – 4x + 4 is a product. Factoring polynomial expressions that are not perfect squares are more difficult. The best way to figure them out is to figure out the factors of the whole number.
  x2 – 5x – 36 The factors of 36 are {1, 2, 4, 6, 9, 12, 18, 36

Every factor has a pair: 1 and 36, 4 and 9, 6 and 6 (the square root). Add or subtract these numbers to find a number that equals 5 (the second number in the original question).

The solution is (x - 9)(x + 4) because -9x + 4 = -5

To solve an algebraic equation using polynomials, set the polynomial equal to 0 and solve for all the possible values of x.

        x2 – x + 10 = 12
x2 - x + 10 - 12 = 12 - 12
         x2 - x - 2 = 0
    (x + 1)(x - 2) = 0

x + 1 = 0 x - 2 = 0
      x = -1      x = 2
             x = -1 or 2


By Matthew DeFour


Worksheet:

  1. Find the value for f(3):
    1. f(x) = 3x
    2. f(x) = 4x2 – 5x + 2
    3. f(x) = 9x4 + 32x2 – 40x + 1
    4. f(x) = 64x2y – 32xy + 4y
  2. Solve for x
    1. 3x – 4 = 11
    2. 22x + 41 = 5x – 10
    3. 4x + 3x + 2 = 37
    4. 4(3x – 2) = 40
  3. Find the product
    1. (x – 2)(3x + 5)
    2. (y + 10)(y – 10)
    3. (4x + 5)(2x – 1)
    4. (3x2 – 1)(4x + 1)
  4. Factor
    1. 3x2 – 15x + 18
    2. x2 – 10x + 24
    3. 16y2 + 24y – 16
    4. 54y2 + 30y – 4
  5. Solve for x
    1. x2 – 6x + 12 = 3
    2. 4x2 + 5x – 7 = 5x2 – 6x + 23
    3. x2 – 2x + 64 = x2 + 4x – 2
    4. x3 – 27 = 0


Answers:
1a. 9
1b. 23
1c. 898
1d. 484y
2a. x = 5
2b. x = -3
2c. x = 5
2d. x = 4
3a. 3x2 – x – 10
3b. y2 – 100
3c. 8x2 + 6x – 5
3d. 12x3 + 3x2 – 4x – 1
4a. 3[(x – 3)(x – 2)]
4b. (x – 6)(x – 4)
4c. 8[(2y – 1)(y + 2)]
4d. (6y + 4)(9y – 1)
5a. x = 3, -3
5b. x = 5, 6
5c. x = 11
5d. x = 3