Polynomials and Equations

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Polynomials and Equations

Exponent Laws

Power of Power Rule

Ex. (xᵃ)ᵇ= xᵃˣᵇ

Quotient Rule

Ex. xᵃ÷xᵇ= xᵃ⁻ᵇ

Product Rule

Ex. xᵃ×xᵇ= xᵃ⁺ᵇ

w

Ex. 7×7×7=7³

base=7

Exponent= ³

Algebraic expression

Modelling Tiles

Variable

7x+3

Number, variables, operators

distributive property

Simplifies complicated expressions

Example:

Subtopic

Term

4x²

Coefficient= 4

Varriable= x²

degree of a term

Sum of exponents

2/3xy= 1+1=2

Like terms

identical variables with same exponents on each variable

Subtract: 8x-3x =5x

Add: 4x+ 3x = 7x

Polynomial

To subtract, add its opposite: (3y+5)-(7y-4) =(3y+5)+(-7y+4) =3y+5-7y+4 =3y-7y+5+4 =-4y+9

To add, remove bracket, collect like terms: (2p-2)+(4p-7) = 2p-2+4p-7 = 2p+4p-2-7 = 6p-9

Made up of term(s) connected by addition or subtraction operators.

Expression VS Equation

Has an equal sign Fractions can be removed (by finding LCD) We can solve

Does not have an equal sign Can't remove fraction (only expand) We can only simplify

Modelling with Algebra

Example: Alexa works at a record shop. She earns $!0.70/hr plus $0.88 for each album she sells. To model this situation we can say, "h " is the number of hours worked and "a" is the number of albums sold. The expression would be 10.70 h + 0.88 a

It is a representation of a pattern of numbers.

Modelling with Formulas

To rearrange, isolate the term that contains the variable, and then isolate the variable.

Algebraic relationship between two or more variables.

Equations with fractions

For more than one fraction, find the LCD and multiply all the terms on both sides of equation by this value.

k + 2/3 = k - 4/ 5 15 ✕ k + 2/3 = 15 ✕k - 4/ 5 5 (k+2) = 3 (k-4) 5k + 10 = 3k - 12 5k + 10 - 3k - 10 = 3k - 12 - 3k - 10 2k = - 22 2k/2 = - 22/2 k = - 11

Eliminate the fraction by multiplying both side of the equation by the denominator.

6 = ⅓ (8+x) 3 ✕ 6 = 3 ✕ ⅓ (8+x) 18 = 8 + x 18 - 8 = 8 + x - 8 10 = x

Multi-Step Equations

To solve equations with brackets, you need to expand them. 5 (y-3) - (y-2) = 19 5 (y-3) - (y-2) = 19 5y - 15 - y +2 = 19 4y - 13 = 19 4y - 13 + 13 = 19 4y = 32 4y/4 = 32/4 y=8

Check: L.S.= 5 (y-3) - (y-2) R.S.= 19 =5y - 15 - y +2 = 5 (8) - 15 - (8) + 2 =40 -15 -6 = 19 L.S.= R.S.

To solve an equation with Multiple Terms, collect variable terms one side and constant* terms on the other by doing opposite operation. Ex: 3x +2 = 2x -4 3x +2 -2x = 2x -4 -2x x +2 = -4 x +2 -2 = -4 -2 x = -6

Simple Equations

You can also do a check: L.S. = 2(8) - 7 R.S. = 9 =16 - 7 = 9 L.S. = R.S.

To solve a 2 step equation, either add or subtract to isolate the variable term. Then divide by the coefficient of the variable term. Ex: 2x -7 = 9 2x -7 +7 = 9 +7 2x = 16 2x/2 = 16/2 x=8 Keep in mind that you are using SAMDEB

To solve one step equation, do opposite operation to isolate the variable. Ex: x+4=13 x +4 -4 = 13 -4 x=9

Equation is a statement that say 2 expressions are equal. Ex: 3x+3=2x-1

Solution or root of an equation is the value of the variable that makes an equation true.