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Bell Quiz

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Objectives Learn to simplify expressions using the greatest common factor or GCF.

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Prime Factorization Simplifying expression that contain numbers often requires knowledge of prime number and factors. Recall that a prime number is a whole number that is only divisible by itself and 1. All whole numbers other than 1 that are not prime are composite numbers. Composite numbers have whole-number factors other than 1 and the number itself. They can be written as a product of prime numbers, which is called prime factorization.

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Prime Factorization Several methods can be used to find the prime factorization of a number. The process requires breaking down the composite umbers until all the factors are prime. The Prime factorization for number 24 can be found in at least three ways.

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**Prime Factorization Example**

The Prime factorization for number 24 can be found in at least three ways. It does not matter which method is used to find a prime factorization.

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**Example 1 Finding the Prime Factorization of a Number**

Find the prime factorization of the number. 120

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**Example 2 Finding the Prime Factorization of a Number**

Find the prime factorization of the number. 924

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Lesson Practice Find the prime factorization of the number. 100

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Lesson Practice Find the prime factorization of the number. 51

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**Greatest Common Factor**

Prime factorization can be used when determining the Greatest Common Factor (GFC) of Monomial. The GFC of a monomial is the product of the greatest integer that divides without remainder into the coefficients and the greatest power of each variable that divides without a remainder into each term. Finding the GCF means finding the larges monomial that divides without a remainder into each term of a polynomial.

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**Example 3 Determining the GCF of Algebraic Expession**

Find the GCF of the expression. 6a2b3 + 8a2b2c

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**Example 4 Determining the GCF of Algebraic Expression**

Find the GCF of the expression. 24m3n4 + 32mn5p

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Lesson Practice Find the GCF of the expression. 8c4d2e – 12c3d4e2

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Lesson Practice Find the GCF of the expression. 5p2q5r2 – 10 pq2r2

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**Factoring Polynomials**

Finding the GCF of a polynomial allows you to factor it and to write the polynomial as a product of factors instead of the sum or difference of monomials. Factoring a polynomial is the inverse of the Distributive Property. Using the Distributive Property will “undo” the factoring of the GCF.

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**Example 5 Factoring a Polynomial**

Factor the polynomial completely. 6x3 + 8x2 – 2x

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Lesson Practice Factor the polynomial completely. 8d2e3 + 12d3e2

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**Example 6 Factoring a Polynomial**

Factor the polynomial completely. 9x4y2 – 9x6y

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Lesson Practice Factor the polynomial completely. 12x4y2z – 42x3y3z2

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Algebraic Fractions Fractions can be simplified if the numerator and denominator contain common factors. This is because the operations of multiplication and division undo each other. An algebraic fraction can only be simplified if the numerator and the denominator have common factors.

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**Example 7 Simplifying Algebraic Fractions**

Simplify the expression 3p + 3 3

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Lesson Practice Simplify the expression 6x + 18 6

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**Example 8 Simplifying Algebraic Fractions**

Simplify the expression 5x – 25x2 5xy

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Lesson Practice Simplify the expression 18x + 45x3 9x

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**Example 9 Application: Finding the Height of an Object**

The formula h = –16t2 + 72t + 12 can be used to represent the height of an object that is launched into the air from 12 feet off the ground with an initial velocity of 72 feet/second. Rewrite the formula by factoring the right side using the GCF and making the t2–term positive.

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Lesson Practice The formula h = –16t2 + 60t + 4 can be used to find the height of an object that is launched into the air from 4 feet off the ground with an initial velocity of 60 feet/second. Rewrite the formula by factoring the right side using the GCF and making the t2–term positive.

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Multiplying and Factoring

Multiplying and Factoring

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