Sections 7.2, 7.3, 4.1
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1 Sections 7., 7.3, 4.1 Section 7. Multiplying, Dividing and Simplifying Radicals This section will discuss the rules for multiplying, dividing and simplifying radicals. Product Rule for multiplying radicals For non- negative real numbers a and b, a b = ab, and ab = a b Examples: r, r 0 Simplifying Radicals using the product rule A square root radical is simplified when no perfect square factor larger than 1 remains under the radical sign. Examples: Find the product and simplify
2 Simplifying Radicals Using the Quotient Rule If a and b are nonnegative real numbers and b 0 then a a a a = and = b b b b Use the quotient rule for simplifying radicals Using both rules together Simplifying Radicals Involving Variables Since a means the principal square root, the result is a positive number, and a is the negative square root, so its result is negative.
3 So for a, regardless of whether a was a positive or negative number, the fact that the square root has no negative sign means that the result is positive, and the result of a is negative. Why does this matter? = 4 = ( ) = 4 = The order of operations would have you square the number under the radical first, then take its principal square root. Regardless of whether the original number being squared was positive or negative the principal square root of it is always positive. To write that in variables: a = a If we assume that a 0, then a = a so the absolute value bars are unnecessary. We will assume that the variables represent positive numbers. If the exponent on the variable is even, 1. 8 x n a is found by having the exponent y n Let s look at an example of what to do when the exponent is odd: 5 a So you have to use the multiplication property of radicals, and simplify the part with the even square root, and leave the extra radical a
4 Simplifying other roots For cube roots, look for factors that are perfect cubes. For variables, divide by 3 Other roots work similarly m Section 7.3 Solving Equations with Radicals To solve equations involving radicals, you will have to use what is called the squaring property of equality. If an equation has a square root on one side of the equation, you can square both sides of the equation. And since ( a ) = a this is a very helpful property! 1. k = 3. x = x = 0x + 5
5 4. x = x 4x 16 Section 4.1 Adding and Subtracting Polynomials In this section we will discuss: Review combining like terms Know the vocabulary of polynomials Evaluate polynomials Add polynomials Subtract Polynomials Add and subtract polynomials with more than one variable Definitions: Terms: an expression such as 9x x + 3x 8 is an expression that has 4 terms. Terms are separated by plus or minus signs, but the sign is 4 considered associated with the term. So the terms are9x, x,3x, 8 The numbers at the beginning of the terms are called coefficients. The coefficients are 9,,3, 8. The -8 is called the constant term. Like terms and Unlike terms: Like terms have exactly the same combination of variables with the same exponent on the variables with the same exponents on the variables. Only the coefficient may differ. Unlike terms have different variables or different exponents on the same variables. Examples of like terms: Examples of unlike terms:
6 Polynomials: A polynomial in x is a term or the sum of a finite number of terms of the form n ax for any a and any whole number, n. What does this mean?: if you are adding or subtracting a bunch of terms of a particular type, you have what is called a polynomial. What makes a polynomial a polynomial? It is the fact that all the terms have exponents that are whole numbers, which really means that no exponents are fractions AND no variables appear in denominators. Polynomials are typically written in descending order of powers. 4 This is a polynomial: 9x x + 3x 8 But this is not a polynomial: 9x x + 3 x Degree of a term: The degree of a term is the sum of the exponents on 4 the variables. A constant term has degree 0. For example, 9x has degree 4, 4 6 and 9x y has degree 10 (b/c = 10) Degree of a polynomial: The degree of a polynomial is the greatest 4 degree of any nonzero term of the polynomial. For example 9x x + 3x 8, the degree is 4. Names of different types of polynomials: Monomial: a single term. For example: 4 9x 4 1 Binomial: Contains two terms: 4x x Trinomial: Contains three terms: 8x 7 x 8 Examples of simplifying polynomials:
7 Simplify each polynomial if possible, then give the degree and tell whether the polynomial is a monomial, binomial, trinomial, or none of the above. 1. 3x + x x + 4x x x + x Evaluating Polynomials This means to find the value the polynomial calculates out to be after substituting a number (or numbers) for the variable(s). Find the value of x 3 + 8x 6in each case: 1.when x = -1. when x = 4 Adding Polynomials: To add polynomials, add like term. Add the following polynomials: 3 1. ( 4x 3x + x)+ 6x + x 3x 3 ( )
8 Subtracting Polynomials: To subtract two polynomials, change all of the signs of the second polynomial and add the result to the first polynomial. You can think of the polynomial after the minus sign as having a -1 coefficient outside the parentheses. You can then distribute the -1 through, thus changing all the signs. For example: 3 14y 6y + y ( ) ( y 7y 4y + 6) Examples: 3 4y 16 y 3 1. ( + y) ( 1 y 9y + 16). ( 7 xy 6x y + y) ( xy 7y + 5x y)
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