MAT 120/121 Chapter 4 When you multiply like bases you add the exponents. When you divide like bases you subtract the exponents.

A non-zero base to the power of 0 = 1. A power to a power, multiply the powers.

Simplify by using all exponential properties and positive exponents.

MAT 120/121 Chapter 4 Negative exponents moves the base from the top to the bottom or bottom to the top. Write using positive exponents. Simplify, if possible.

Use exponent properties and write using positive exponents Use exponent properties and write using positive exponents. Simplify, if possible.

MAT 120/121 Chapter 4 Place the decimal point to create a number between 1 and 10. Count the number of decimal places, this is the power on 10. Notice that the power is negative! Exponent sign rule. Original number > 1, Positive exponent. Original number < 1, Negative exponent.

Place the decimal point to create a number between 1 and 10. Count the number of decimal places, this is the power on 10.

MAT 120/121 Chapter 4 Multiply like bases of 10, add the exponents Change 62.9 to scientific notation. Multiply the two decimal numbers. TOO BIG TOO SMALL Divide like bases of 10, subtract the exponents Change 0.25 to scientific notation. Divide the two decimal numbers.

(quantity) MAT 120/121 Chapter 4 number variable multiplied divided sum difference divided A one term polynomial. 3x2 7x 2xyz 6 2xyz – 6 A two term polynomial. 3x2 + 7x 3x2 + 7x – 6 A three term polynomial.

The sum of the powers on the variables in one term. 2 + 3 = 5, 5th degree The term with the highest degree will be the degree of the polynomial. The value that is multiplied to a variable in a term. The term with the highest degree will be the leading term of the polynomial ONLY if the polynomial consists of one variable! The value that is multiplied to the leading term. Polynomials that consists of one variable are traditionally written with the powers on the variable in descending order. This way the leading term is written first in the polynomial. Below 3x2 is the leading term, 3 is the leading coefficient, and the polynomial is a 2nd degree. 3x2 + 7x – 6

Write the polynomial in descending order.

( ) ( ) ( ) ( ) MAT 120/121 Chapter 4 powers/exponents Largest powers 1st ( ) ( ) ( ) ( ) ( )’s around every variable x and the value you want to substitute. Place the -2 into the ( )’s Evaluate each term separately and then combine all the values. OPPOSITES cancel!

Combine Like Terms! Notice that the polynomials are written in descending order! This will make C.L.T. much easier!

MAT 120/121 Chapter 4

5x 3x 15 x*x = x2 x2 + 5x + 3x + 15 Area = x2 + 8x + 15 sq. units Area is the space inside the rectangles. Find the area of each rectangle. A = L * W Perimeter is the distance around the object. Fill in the missing outer sides and add to find the sum of the outer sides. These are all rectangles so opposite sides are the same. x 5 Add each individual area together. x x*x = x2 5x x2 + 5x + 3x + 15 3 3x 15 Area = x2 + 8x + 15 sq. units x + x + x + x + 3 + 3 + 5 + 5 Perimeter = 4 x + 16 units Area of a circle is r = x The area will be BIG CIRCLE – small circle. r = 5 sq. units sq. units

MAT 120/121 Chapter 4 Multiply like parts. Re-group by Comm. Prop of Mult. Distributive Property

Double Distributive Property or F.O.I.L. F = First terms O = Outer terms I = Inner terms L = Last terms Double Distributive Property or F.O.I.L. F O I L x2 – 3x + 6x – 18 3x3 – 12x2 + 5x – 20 C. L .T No like terms 4x2 – 9 x2 – 6x + 6x – 36 Cancels Binomials with the same 1st terms, but opposite 2nd terms are called CONJUGATE PAIRS. The middle terms always cancel. ( a + b ) ( a – b ) = a2 – b2. Just multiply F and L. “2x * 2x” “5 * 5” * * * Dble IT Binomials that are squared need to written twice and FOILed. “2x * 5 = 10x Dble IT” Do you see a pattern? Do see the SMILE? The middle terms are always the SAME. When we combine them, it DOUBLES. ( a + b )2 = a2 + 2ab + b2 or ( a – b )2 = a2 – 2ab + b2. Mr. Fitz does the SMILE technique. SMILE? SMILE? SMILE?

4x3 – 5x2 – 3x + 8x2 – 10x – 6 We will need more arrows. No room for more arrows! Try vertical alignment. I prefer the polynomial with coefficients of 1 on the bottom. Line up like terms 4x3 – 5x2 – 3x C.L.T. + 8x2 – 10x – 6

( ) ( ) ( ) ( ) ( ) ( ) ( ) MAT 120/121 Chapter 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )’s around all variables and the value you want to substitute. Evaluate each term separately and then combine all the values. Because the powers are the same, we can move the -2 and 5 into the ( )’s and make it -10 which is easier to multiply by! If the directions want the answer in decimal form, use and not the symbol on the calculator. The answer will be to big. If the directions want the answer in terms of pi, treat pi like a variable and leave it in the answer.

Find the degree of the terms and the polynomial. It will help to keep variables in alphabetical order to better recognize the same powers on the variables Combine like terms. 1 Find the degree of the terms and the polynomial. 1 1 1 1 0 deg. 5th 6th 2nd 1st 2nd The degree of the polynomial is a 6th degree. Perform the indicated operations. Distribute the minus sign!

Perform the indicated operations. 2x2 – 3xy + 10xy – 15y2 C. L .T * * * Dble IT If you look close, these two trinomials can be re-written as a conjugate pair of binomials. “2x * 2x” “3 * 3” * * * Dble IT “2x * 3 = 6x Dble IT”

3 “GAZINTA” 15, 5 times and subtract exponents.

Polynomial divided by a binomial. Old style LONG DIVISION Our goal is to eliminate the leading term on the inside. Therefore we only divide leading terms. Line up the x above 5x. Distribute to the front terms and line them up under the division box. x2 + 3x Subtract both columns. 2x + 6 Leading terms must CANCEL! Bring down the next term and repeat. New leading terms to divide. Line up the 2 above 6 as + 2. Distribute to the front terms and line them up under the division box. Subtract both columns.

Polynomial divided by a binomial. Our goal is to eliminate the leading term on the inside. Therefore we only divide leading terms. Line up the x above 5x. Distribute to the front terms and line them up under the division box. 2x2 – 1x Subtract both columns. Fix double signs. 6x – 3 Leading terms must CANCEL! Bring down the next term and repeat. New leading terms to divide. We have a remainder. We will add it to the answer as plus a fraction. Line up the 3 above 1 as + 3. Distribute to the front terms and line them up under the division box. Subtract both columns.

Polynomial divided by a binomial. In this problem, notice that the x3 + 1 is missing terms. We must put in 0’s to hold the place value of the terms so we can line them up when subtracting. Divide leading terms. Line up the x2 above 0x2. Distribute to the front terms and line them up under the division box. x3 + 1x2 Subtract both columns. Leading terms must CANCEL! – 1x2 – 1x Bring down the next term and repeat. New leading terms to divide. Line up the – 1x above 0x. Distribute to the front terms and line them up under the division box. Subtract both columns. Bring down the next term and repeat.

Polynomial divided by a binomial.