Problem 24 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ 40 x^{7} y^{2} z, 64 x^{9} y $$

Step-by-Step Solution

Answer

The GCF is \( 8x^7y \).

Step 1: Understanding the Problem

We need to find the greatest common factor (GCF) of the terms in the list: \( 40x^7y^2z \) and \( 64x^9y \). The GCF is the largest expression that divides each term without a remainder.

Step 2: Prime Factorization of the Coefficients

First, let's find the prime factorization of the coefficients: - 40 can be factored into prime numbers as \( 2^3 \times 5 \).- 64 can be factored into prime numbers as \( 2^6 \).The GCF of the numerical coefficients is the product of the smallest powers of common prime factors: - For 2, the smallest power is \( 2^3 \).- 5 does not appear in both numbers.

Step 3: Identifying Common Variable Factors

Now, compare the variable parts of the terms: - For \( x \), the smallest power is \( x^7 \) (comparing \( x^7 \) and \( x^9 \)).- For \( y \), the smallest power is \( y \) (comparing \( y^2 \) and \( y \)).- \( z \) is only present in the first term, so it is not included in the GCF.

Step 4: Writing the GCF Expression

Combine the GCFs of the numerical coefficients and the variables: - Coefficients: \( 2^3 = 8 \).- Variables: \( x^7 y \).Thus, the GCF of \( 40x^7y^2z \) and \( 64x^9y \) is \( 8x^7y \).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into the product of its prime numbers. For example, if you have the number 40, you begin by dividing it by the smallest prime number, which is 2.
- Thus, 40 becomes 20 when divided by 2, and again 20 becomes 10 when divided by 2. - Finally, 10 becomes 5 when divided by 2, and 5 is a prime number. So, the prime factorization of 40 is written as \( 2^3 \times 5 \). Similarly, we do the same for 64:- Start by dividing by 2 first, several times until only prime numbers are left.- 64 becomes 32 by 2, then 16, 8, 4, and finally 2. This makes the prime factorization for 64 to be \( 2^6 \). During GCF finding, we look for the smallest power of common primes. Trace each step and it becomes simple!

Exponents

Exponents are a shorthand way to show repeated multiplication of the same factor. When you see something like \( x^7 \), it means you multiply \( x \) by itself 7 times. It's a powerful way to manage large calculations without writing or computing each factor individually.In our exercise, identifying the smallest exponent in each matching term is key. This rule holds when determining the greatest common factor:

• For variables like \( x \) and \( y \), compare exponents: Choose \( x^7 \) for the smallest power, not \( x^9 \).
• Do the same for \( y \), choosing \( y \) instead of \( y^2 \) as it's smaller.

Combining the smallest exponents with the numerical GCF gives you a perfect and complete answer.

Polynomial Expressions

Polynomial expressions are combinations of variables (like \( x \), \( y \)) and coefficients that form multiple terms. They might look complex, but understanding their structure simplifies everything. A polynomial expression like \( 40x^7y^2z \) blends constants, variables, and exponents.When solving for the greatest common factor in polynomials, dissect each term into its pieces:
- Separate coefficients from variables.- For variables, compare powers (exponents) and pick the smallest ones common to both terms.- In \( 40x^7y^2z \) and \( 64x^9y \), the greatest common factor focuses on traits they share: \( 8x^7y \). Always recognize these elements and maneuver through them. This is at the heart of working with polynomial expressions.