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

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.