1. Exponents and Polynomials
Chapter 7
Exponents and Polynomials

2. Lesson 7-1 Integer Exponents
Zero Exponent – Any nonzero number raised to the zero power is 1.
Ex: 30 = = 1 (-16)0 =1
Negative Exponent – A nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the positive exponent.
Ex: = 1/32 = 1/ = 1/24 =1/ x-n = 1/xn

3. Lesson 7-2 Powers of 10 and Scientific Notation
Powers of 10 and scientific notation can be used to write very large and very small numbers.
Scientific Notation – a number between 1 and 10 times a power of 10 is written in scientific notation.
The exponent on the power of 10 tells how many zeros it contains. With negative exponents, the number of zeros is one less than the exponent number.

4. Lesson 7-2 (cont.)
When working with scientific notation a negative exponent corresponds with moving the decimal point to the left, a positive exponent corresponds with moving the decimal point to the right.
When multiplying large numbers, write them in scientific notation and combine powers of 10.
Ex: (1.43 x 105)(2.6 x 104) =
(1.43 x 2.6)(105 x 104) = x 109

5. Lesson 7-3 Multiplication Properties of Exponents
An exponential expression is completely simplified if…
There are no negative exponents.
The same base does not appear more than once in a product or quotient.
No powers are raised to powers.
No products are raised to powers.
No quotients are raised to powers.
Numerical coefficients in a quotient do not have any common factors other than 1.

6. Lesson 7-3 Multiplication Properties of Exponents
Product of Powers Property –
aman = am+n
Power of a Power Property –
(am) nn = amn
Power of a Product Property –
(ab)n = anbn

7. Lesson 7-4 Division Properties of Exponents
Quotient of Powers Property
am/an = am-n
Positive Power of a Quotient Property
(a/b)n = an/bn
Negative Power of a Quotient Property
(a/b)-n = (b/a)n = bn/an

8. Lesson 7-4 Division Properties of Exponents
Numbers that are very large or small can be simplified using scientific notation:
(4.7 x 10-3) x (9.4 x 103) =
(4.7 / 9.4) x (10-3/103) =
0.5 x 10-6 = 5 x 10 -7

9. Lesson 7-5 Polynomials
Standard form of a polynomial – written with the terms in order from greatest degree to least degree. The coefficient of the first term is called the leading coefficient.
Ex: 18y5 – 3y8 + 14y becomes
-3y8 + 18y5 + 14y
(The leading coefficient is -3).

10. Polynomials can be classified according to their degree and number of terms (see table for ex.)
by degree
by # terms 8
Constant
Monomial
X-2 1
Linear
Binomial
4c2 + c – 3 2
Quadratic
Trinomial
X3+2x2+3 3
Cubic

11. Lesson 7-6 Adding and Subtracting Polynomials
To add or subtract polynomials combine like terms. This can be done in horizontal or vertical form.
To subtract polynomials, remember that you can add the opposite. To find the opposite of a polynomial, you must write the opposite of each term in the polynomial.

12. Lesson 7-6 (cont). Ex: (2x2 – 3x + 1) – (x2 + x + 1)

13. Lesson 7-7 Multiplying Polynomials
To multiply a polynomial by a monomial, use the Distributive Property.
To multiply a binomial by a binomial you can use the distributive property twice, or use the FOIL method.
First terms, Outer terms, Inner terms,
Last terms
Ex: (x + 3)(x + 5) = x2 + 3x + 5x + 15
= x2 + 8x + 15

14. Lesson 7-7 (cont.)
To multiply polynomials with more than two terms, use the Distributive property more than once.
You can also use a rectangle model to multiply polynomials with two or more terms.
(See the textbook or the homework help website for examples).

15. Lesson 7-8 Special Products of Polynomials
Perfect square trinomial a trinomial that is the result of squaring a binomial.
Ex: (x + 6)2 = (x + 6)(x + 6)
= x2 + 6x + 6x + 36
= x2 + 12x + 36
(x - 6)2 = (x - 6)(x - 6)
= x2 - 6x - 6x + 36
= x2 - 12x + 36

16. Lesson 7-8 (cont). Difference of two squares – a binomial of the form
(a+b)(a-b) = a2 – b2
Ex: (x+6)(x-6) = x2 – 6x + 6x -36
= x2 – 36