Exponents and Polynomials Chapter 7 Exponents and Polynomials

Lesson 7-1 Integer Exponents Zero Exponent – Any nonzero number raised to the zero power is 1. Ex: 30 = 1 1230 = 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: - 3-2 = 1/32 = 1/9 2-4 = 1/24 =1/16 x-n = 1/xn

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.

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) = 3.718 x 109

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.

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

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

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

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).

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

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.

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

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

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). http://go.hrw.com/gopages/ma/alg1_07.html

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

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