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Algebra II – Additional Topics

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    Study Guide Algebra II – Additional Topics 1. Factorials What is a Factorial? A factorial is a useful way to write the product of a number multiplied by all the natural numbers that come before it. We use an exclamation mark (!) to represent factorial . For example: 6! is read as “six factorial.” It means: So, Factorials are often used in mathematics, probability, and combinatorics when working with arrangements and counting problems. General Formula for Factorials For any natural number (n): This means we multiply the number by every positive integer that comes before it until we reach 1 .

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    Study Guide What About 0!? You might wonder what happens when we take the factorial of zero . Since 0 is not a natural counting number , the usual multiplication pattern cannot continue. However, in mathematics we define : This definition is important because it helps many mathematical formulas work correctly. Example 1 Evaluate: First, expand the factorials. Now substitute them into the expression: Next, cancel the common factors in the numerator and denominator. After simplifying, we get:

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    Study Guide 2. Quiz: Factorials 1. Question Find the value of 7! Answer Choices • 5040 • 6050 • 7060 Correct Answer : 5040 Why This Is Correct The factorial symbol (!) means multiplying a number by all positive integers smaller than it. Step by step: 7 × 6 = 42 42 × 5 = 210 210 × 4 = 840 840 × 3 = 2520 2520 × 2 = 5040 So, [7! = 5040]

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    Study Guide 2. Question Find the value of 8! / 6! Answer Choices • 54 • 56 • 58 Correct Answer : 56 Why This Is Correct Write the factorials in expanded form: Now substitute into the fraction: The 6! cancels out : So, the correct answer is 56 . 3. Question Find the value of (8! × 3!) / 9!

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    Study Guide Answer Choices • 7/8 • 3/4 • 2/3 Correct Answer : 2/3 Why This Is Correct First expand the factorials. Now substitute: Simplify: So the correct answer is 2/3 . 4. Question Find the value of (5! × 7!) / (3! × 6!)

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    Study Guide Answer Choices • 80 • 110 • 140 Correct Answer : 140 Why This Is Correct Expand the factorials: 5! = 120 7! = 5040 3! = 6 6! = 720 Substitute into the expression: Multiply: Now divide:

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    Study Guide So the correct answer is 140 . 5. Question Find the value of (8! × 9!) / (6! × 7!) Answer Choices • 3012 • 4032 • 5052 Correct Answer : 4032 Why This Is Correct Rewrite the factorials to simplify: Substitute into the expression: Cancel common factorial terms: After cancelling 6! and 7 , we get:

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    Study Guide So the correct answer is 4032 . 3. Binomial Coefficients and the Binomial Theorem Understanding Binomial Expansions When a binomial expression (an expression with two terms) is raised to a whole number power, the result follows a clear and interesting pattern. Consider the expression: Let’s look at the first few expansions. Notice that the numbers in front of each term (the coefficients) follow a pattern. Patterns in Binomial Expansions From the examples above, we can observe several important patterns. 1. Number of Terms

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    Study Guide Each expansion has one more term than the exponent . Example: • Power 3 → 4 terms • Power 4 → 5 terms 2. Sum of the Exponents In every term, the sum of the exponents of (a) and (b) equals the power of the binomial. Example for ((a+b) 4 ): Each term has exponents that add up to 4 . 3. Changing Powers As we move from left to right: • The power of (a) decreases by 1 • The power of (b) increases by 1 4. Symmetrical Coefficients The coefficients form a symmetrical pattern . Example: They read the same forward and backward . 5. Pascal’s Triangle Pattern

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    Study Guide Each number in the expansion can be obtained by adding the two numbers above it . This triangular arrangement of numbers is called Pascal’s Triangle , named after the French mathematician Blaise Pascal . Pascal’s Triangle Pascal’s Triangle helps us find the coefficients when expanding binomials. Example rows: These numbers are exactly the coefficients used in binomial expansions . Using Factorials to Express the Coefficients The numbers in Pascal’s Triangle can also be written using factorials . For example: This expression gives the binomial coefficient . It tells us the coefficient of a particular term in the expansion. The notation used is:

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