If you cube one side of an equation, you must cube the other. You get the idea. As you review the following properties of addition and multiplication, remember these are the properties that allow you to perform operations needed to simplify expressions and solve equations. It is important to be able to perform these operations and identify what property is being used. Assume a, b, and c are real numbers.
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Properties of Addition 1. It is also called the additive inverse of a. Properties of Multiplication 1. It is also called the multiplicative inverse a of a. Distribute the 2. Answer The example shows that the distributive property also works for trinomials expressions containing three terms. Does it apply to multiplication over subtraction? Try the next example to see that it does hold true. Distribute the 5. Answer Properties of Positive and Negative Numbers You should be familiar with working with signed numbers by this point in your math career.
Here is a review of the basic properties of positive and negative numbers. This is a variation of the property of opposites that usually tricks students. It is often used when simplifying rational expressions and factoring, so it is important to recognize. Substitute the given values. Answer To solve an equation involving absolute value, think in terms of two separate equations: one where the expression inside the absolute value signs is positive and one where the expression inside the absolute value is negative. It is important to recognize immediately that y will never be negative because it equals the absolute value of an expression.
Absolute value, by definition, is a positive distance. If n is not written, it is assumed to equal 2, for example 4 read the square root of 4. The positive square root, 4 in this case, is called the principal square root. Two solutions One solution It is important to note the distinction in the examples above. In Example 1, it is true that 49 has two real roots, but 49 has only one solution.
This is always a good idea! Simplest Radical Form Make sure two things are true when writing an expression, such as simplest radical form: n x , in 1. Factor all perfect nth powers from the radicand, and 2.
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Rationalize the denominators so that no radicals remain in the denominator and no radicands are fractions. Write the following examples in simplest radical form. Rationalizing the Denominator To rationalize a denominator containing a radical, try to create a perfect square, cube, or other nth power. You must multiply by a perfect square this time in order to create a perfect cube in the denominator. Multiplying the denominator by 22 will create a perfect cube, 2 22 22 3 2 , in the denominator.
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Distribute 3. The denominator is rationalized here, but the numerator is 3 not in simplest radical form. Assuming a and b are integers, the product of conjugates will always equal an integer. Conjugates are useful when rationalizing a denominator containing a binomial radical expression. Now simplify. A polynomial contains many terms. Polynomials can be added, subtracted, multiplied, and divided following the properties of real numbers.
Adding and Subtracting Polynomials Polynomials can be added and subtracted by combining like terms. Like terms have the same variables raised to the same power. In other words, they are terms that differ only by their coefficients. Set up your expression. Subtract the first term from the second. Change the subtraction to adding the opposites. Combine like terms. You are probably most familiar with multiplying a binomial a polynomial with 2 terms by another binomial. Answer Remembering these special products of polynomials will help when factoring. Factoring means expressing a polynomial as a product of other polynomials.
The most basic way to factor a polynomial is to distribute out its greatest monomial factor. Remember, a monomial is a single term, such as a constant, a variable, or the product of constants and variables.
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The GCF of the terms is xy. Trinomials Some trinomials can be factored by recognizing that they are a special product. These are called perfect square trinomials. Recognize that half of the second term squared 1 equals the third term.
For these problems, try thinking about the FOIL method, and use trial and error. Think about the factors of the last term. What two numbers will multiply to give you 22 and add to give you 13? Multiply the binomials to check your work.
Because the three terms in the original trinomial are positive, both binomials contain positive terms. Think about the First terms. Check your work. Pay special attention to the positive and negative signs. Difference of Perfect Squares The product of the sum of two terms and the difference of the same terms is called the difference of perfect squares. When factoring, it is important to recognize that this is a special product. Remembering this will save time when factoring. Always factor out the greatest common factor first. Here the GCF is x. Sum and Difference Of Cubes Factoring the sum and difference of cubes is not obvious.
None of the four, however, give you a positive 9 constant and a negative 8 coefficient for x. The solutions are called roots of the quadratic, values of the variable that satisfy the equation.