Factoring out the GCF

Factoring out the GCF exercise

• Describe the greatest common factor (GCF).

  Hints

  To determine the GCF, let's look at an example:

  $6 = 2 \times 3$ and $8 = 2 \times 2 \times 2$

  Both numbers, $6$ and $8$, have the factor $2$ in common, so $2$ is the GCF.

  Now let's take a look at another example:

  $12 = 2 \times 2 \times 3$ and $8 = 2 \times 2 \times 2$

  As you can see, both numbers have the factor 2 listed twice. So the GCF is $2 \times 2 = 4$.

  Solution

  Wow, Gi Na, Chiako, and Fernando have a lot of hobbies. They decide to spend time together and want to do an activity they all enjoy - a common interest.

  The common interest in this example corresponds to the Greatest Common Factor - GCF.

  Let's have a look at the list to the right:

  • Gi Na and Fernando love Sudoku puzzles, but Chiako doesn't care for them.
  • However, we can see that everyone likes karaoke.

  So karaoke is the common interest of the three friends

  Mathematically the GCF is defined as follows:

  To find the greatest common factor of a polynomial, find the factors common to each term in the polynomial.

• Determine the greatest common factor.

  Hints

  You can check the results by using the Distributive Property.

  Look at the following example:

  • $6x^2y=(2 \times 3)(x \times x)(y)$
  • $8xy^2 = (2 \times 2 \times 2)(x)(y \times y)$

  As you can see, both terms have the factors $2$, $x$, and $y$ in common.

  Therefore, the GCF is $2xy$.

  To factor a polynomial, write the polynomial as a product where one factor is the GCF, and the other factor is a polynomial listing all the remaining terms.

  For example:

  $6x^2y - 8xy^2 = (2 \times 3)(x \times x)(y) - (2 \times 2\times 2)(x)(y \times y) = 2xy(3x - 4y)$.

  As you can see, both terms have the factors $2$, $x$, and $y$ in common. So the greatest common factor of both terms is $2xy$.

  Factoring out $2xy$ from $6x^2y$ leaves $3x$. And factoring out $2xy$ from $-8xy^2$ leaves $-4y$. Therefore, the remaining terms comprises the polynomial $3x - 4y$.

  Solution

  $10x^3y^2 + 18x^2y - 4x$

  Oh boy, what a funky polynomial.

  To factor out the GCF:

  We write each term as a product:

  • $10x^3y^2 = (2 \times 5)( x \times x \times x)(y \times y)$
  • $18x^2y = (2 \times 3 \times 3)(x \times x)(y)$
  • $4x = (2 \times 2)(x)$

  You can see, that $2$ and $x$ are common factors. So $2x$ is the greatest common factor.

  Now we factor out the GCF to get:

  $10x^3y^2 + 18x^2y - 4x = 2x(5x^2y^2 + 9xy - 2)$

  You can check your solution using the Distributive Property. To do this, multiply $2x$ by each term listed inside the parentheses.

• Find the greatest common factor.

  Hints

  You don't have to determine the sum of the fruits.

  Determine the common factor for each type of fruit.

  Solution

  First, consider how many of each fruit the friends brought for the camping trip.

  Apples

  • $4$ apples from Gi Na
  • $2$ apples from Chiako
  • $2$ apples from Fernando

  Since $2$ is a factor that Chiako and Fernando have in common, we now need to see if the number of apples Gi Na brought also has $2$ as a factor. $4 = 2 \times 2$, So $2$ is a factor of $4$.

  $2$ apples is our greatest common factor.

  Bananas

  • $2$ bananas from Gi Na
  • $4$ bananas from Chiako
  • $6$ bananas from Fernando

  • $2 = 2 \times 1$
  • $4 = 2 \times 2$
  • $6 = 2 \times 3$

  So $2$ bananas is our greatest common factor.

  Kiwis

  • $1$ kiwi from Gi Na
  • $2$ kiwis from Chiako
  • $0$ kiwis from Ferndando

  Since Fernando didn't bring any kiwis and since $0$ does not have any factors, there is no GCF for kiwis.

• Consider each term and find the greatest common factor.

  Hints

  First factor each number or term as much as possible.

  For example. to determine the GCF of $45$ and $60$, factor the numbers:

  • $45 = 3 \times 3\times 3\times 5$
  • $60 = 2 \times 2\times 3\times 5$

  Next check for factors that are common to both numbers. $45$ and $60$ have the factors $3$ and $5$ in common.

  For the final step, multiply the common factors to find the GCF:

  $3 \times 5 = 15$

  Solution

  How can we determine the greatest common factor of two or more given terms?

  To find the greatest common factor of a polynomial, find the factors common to each term in the polynomial.

  If we find more than one factor common to each term, multiply those factors together:

  (1) GCF of $35$ and $77$:

  • $35 = 5 \times 7$
  • $77 = 7 \times 11$

  So the GCF of $35$ and $77$ is $7$.

  $~$

  (2) GCF of $36$ and $24$:

  • $36 = 2 \times 2\times 3\times 3$
  • $24 = 2 \times 2\times 2\times 3$

  The common factors are two $2$s and one $3$. Multiply these factors, and the product is the GCF $2 \times 2 \times 2 = 12$.

  $~$

  (3) GCF of $15x^2$ and $30x$:

  • $15x^2 = 3 \times 5 \times x \times x$
  • $30x = 2 \times 3 \times 5 \times x$

  So the common factors are $3$, $5$, and $x$, so the GCF is the product of these factors: $3 \times 5 \times x = 15x$.

  $~$

  (4) GCF of $15xy$ and $35x^2$:

  • $15xy = 3 \times 5 \times x \times y$
  • $35x^2 = 5 \times 7 \times x \times x$

  The common factors are $5$ and $x$. The GCF is $5 \times x = 5x$.

• Determine the greatest common factor for all the terms listed in an expression.

  Hints

  Remember, you are looking for the greatest common factor, not just any factor.

  Make a list of factors for each term and see which factors are common to all terms.

  The product of the factors common to each term is called the greatest common factor.

  Solution

  First, we factor each term and then look for the common factors:

  $4x^2 + 28x = (2 \times 2) \times (x \times x)+ (2 \times 2 \times 7) \times (x)$

  Now we see that two $2$s and $x$ are factors common to all terms. So the product of $2 \times 2 \times x$ is the greatest common factor, $4x$.

  So when we factor out the GCF from the original expression, we get:

  $4x^2 + 28x = 4x(x + 7)$

  Again, you can check your solution using the Distributive Property. Multiply $4x$ by each term listed inside the parentheses.

• Name the Greatest Common Factor for each given term.

  Hints

  To find the greatest common factor, first determine all the common factors, and then calculate the greatest common factor as the product of the common factors.

  For example:

  Find the GCF of $32xy$ and $36x^2$. First factor both terms to primes:

  • $32xy = 4 \times 8 \times x \times y$
  • $36x^2 = 4 \times 9 \times x \times x$

  As you can see, both terms have the factors $4$ and $x$ in common, so $4 \times x = 4x$ is the greatest common factor.

  Solution

  We can write each number as the product of its prime factors.

  For example, the prime factors of $32x^2$ are $2 \times 2 \times 2 \times 2 \times 2 \times x \times x$.

  (1) Find the GCF for the expression $24x^3-16x^2+8x$.

  First, prime factor each term:

  • $24x^3 = 2 \times 2 \times 2 \times 3 \times x \times x \times x$
  • $16x^2 = 2 \times 2 \times 2 \times 2 \times x \times x$
  • $8x = 2 \times 2 \times 2 \times x$

  We can see that the three terms have three $2$s as well as an $x$ in common,so the product of these common factors: $2 \times 2 \times 2 \times x=8x$ is the GCF. We can factor this out of our original expression as follows:

  $24x^3 - 16x^2 + 8x = 8x(3x^2 - 2x + 1)$

  $~$

  (2) $12x^2+6x-18$

  Factoring each term, we get:

  • $12x^2 = 2 \times 2 \times 3 \times x \times x$
  • $6x = 2 \times 3 \times x$
  • $18 = 2 \times 3 \times 3$

  Here, we have only one common factor, $6$. So this is also our GCF. When we factor $6$ from our original expression, we write:

  $12x^2 + 6x - 18 = 6(2x^2 + x - 3)$

  $~$

  (3) $36x^4 + 18x^2 - 24xy$

  Factoring each term, we list the prime factors:

  • $36x^4 = 2 \times 2 \times 3 \times 3 \times x \times x \times x \times x$
  • $18x^2 = 2 \times 3 \times 3 \times x \times x$
  • $24xy = 2 \times 2 \times \times 2 \times 3 \times x \times y$

  The common factors of all three terms are $6$ and $x$, so the GCF $6x$. Factoring this out of the original expression, we can write:

  $36x^4 + 18x^2 - 24xy = 6x(6x^3 + 3x - 4y)$.

  You can always check your solution using the Distributive Property.