6/01/2010 @ 11:27 PM
Polynomials
In this unit, we learned about polynomials, exponent laws, terms, distributive property,
There are 5 exponent laws, and they are:
Multiplication
Division
Power Law
Power of a Product
Power of a Quotient*Remember, these rules/laws only apply to multiplication and division problems! So these laws cannot be used with addition and subtraction problems.
*Also, the bases must be the same for the laws to work!
EXAMPLE: 3² x 3² = 3^4
Definitions for this unit are found on the sidebar or -HERE-
Also, we learned about coefficients and variables. The coefficient is the number, and the variable is.. well, the variable and/or exponent. For example, k² - the coefficient is 1 (invisible 1 in front of variable) and the variable is k² .
There are different types of polynomials, so to seperate them, we find the degree.
The degree is the sum of exponents on the variables in a term.
If the term is x², the sum of the exponents is 2, so the degree is therefore 2.
If the term is 3x²y, the sum of the exponents is 3, so the degree is therefore 3.
TIP: if the term is -4, the sum of the exponents is 0, so the degree is 0 as well.
Do you see the pattern? So technically the sum of the exponents in the polynomial is the degree.
TIP: The degree of a polynomial is the degree of the highest degree term!
If we look at polynomials, the term with the highest degree will be the degree of the polynomial! For example, if the polynomial is x + 4, the term with the highest degree would be x, and the degree would be first.
If the polynomial is 5x² - 7x, the term with the highest degree is 5x², and the degree of the polynomial would be second, and so on and so on.
TIP: For terms to be alike, they have to have exact variables and exponents, but the coefficients can be different- like 4xy and 3yx.
We also learned about collecting like terms. To collect like terms, simplify the polynomial by adding or subtracting like terms. ONLY add/subtract coefficients! The variables and exponents stay the same!
As well as that, we learned how to add and subtract polynomials, and the distributive property.
To add polynomials, you remove the brackets and collect like terms.
EXAMPLE: (a² - 7a + 4) + (-5a² + 8a + 5)
= a² + 7a + 4 + (-5a²) + 8a + 5
= a² - 7a + 4 - 5a² + 8a + 5
= -4a² + a + 9
Also, we learned how to subtract polynomials. To do this, you would add the opposite polynomial.
EXAMPLE: (3x² + 2) - (5x² + 6)
= (3x² + 2) + (-5x²) - 6 <-- What you do is make every sign on the other side of brackets become the opposite of what it already is- so 3x would be (-3x), -x² would be x², etc. Remember to change the sign in the middle too, or else it will not work!
= 3x² + 2 + (-5x²) - 6
= -2x² -4
Distributive property is something we also learned. Distributive property is used to get rid of brackets in an equation/expression. Any number in front of a bunch of numbers/variables must be multiplied together!
An example would be 2 (x + y). Since brackets usually mean multiplication, multiply them!
So it would be:
2 (x + y)
=2x + 2y
So, 2 (x + y) = 2x + 2y! (This is because they are equal to one another.)
Another example would be: -7 (n-5) + 7 (2n +3)
To solve it, you need to use proper BEDMAS, so multiplication first, then everything else.
-7 (n-5) + 7 (2n +3)
= -7n + 30 + 14n + 21
= 7n + 51
An even trickier question would be this: (x + y) + (2x + 3y)
Since it is x multiplied by x, it would be x².
So,
(x + y) + (2x + 3y)
= 2x² + 3y²
^ This one is trickier because of the exponents, but remember, practice makes perfect!
PRACTICE:
Use the exponent laws.
Answers in RED.
Labels: Lessons
back to the top.