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Bell Quiz
Objectives
• Learn to simplify expressions using the greatest common factor or GCF.
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.
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.
Prime FactorizationExample
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.
Example 1Finding the Prime Factorization of a Number
Find the prime factorization of the number.
120
Example 2Finding the Prime Factorization of a Number
Find the prime factorization of the number.
924
Lesson Practice
Find the prime factorization of the number.
100
Lesson Practice
Find the prime factorization of the number.
51
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.
Example 3Determining the GCF of Algebraic Expession
Find the GCF of the expression.
6a2b3 + 8a2b2c
Example 4Determining the GCF of Algebraic Expression
Find the GCF of the expression.
24m3n4 + 32mn5p
Lesson Practice
Find the GCF of the expression.
8c4d2e – 12c3d4e2
Lesson Practice
Find the GCF of the expression.
5p2q5r2 – 10 pq2r2
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.
Example 5Factoring a Polynomial
Factor the polynomial completely.
6x3 + 8x2 – 2x
Lesson Practice
Factor the polynomial completely.
8d2e3 + 12d3e2
Example 6Factoring a Polynomial
Factor the polynomial completely.
9x4y2 – 9x6y
Lesson Practice
Factor the polynomial completely.
12x4y2z – 42x3y3z2
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.
Example 7Simplifying Algebraic Fractions
Simplify the expression
3p + 3 3
Lesson Practice
Simplify the expression
6x + 18 6
Example 8Simplifying Algebraic Fractions
Simplify the expression
5x – 25x2 5xy
Lesson Practice
Simplify the expression
18x + 45x3 9x
Example 9Application: 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.
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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