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Finding the Greatest Common Factor of a Polynomial

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by looking at factors and terms in polynomials. We'll then review the process of distribution and factoring. We'll talk about factorization with numbers. And finally, we'll do some examples factoring polynomials.

Now, let's look at factors and terms in polynomials. A "polynomial" is an expression involving one or more terms. So here's an example of a polynomial.

7y to the fourth plus 2y minus 3. Terms are separated by addition or subtraction. So in this polynomial, we have three terms.

Each term may contain coefficients, variables, and exponents that are multiplied together. So in our first term, 7 is the coefficient, y is the base, and 4 is the exponent. The coefficients and variables of a term are factors of that term. An expression can also be a factor of a polynomial if it is multiplied by another expression to get the original polynomial as a product. For example, in the expression x minus 1 times 2x plus 5, x minus 1 is a factor and 2x plus 5 is a factor of the polynomial.

Now, let's look at the processes of distribution and factoring. Distribution involves multiplying a factor outside of parentheses by all the terms inside the parentheses. So for example, a times b plus c is equal to a times b plus a times c. The a is multiplied by both the b and the c inside the parentheses.

Factoring is the reverse process of distributing where a common factor is factored out of two or more terms and written outside of the parentheses. So for example, if we start with the expression ab plus ac, a is the common factor of ab and ac. So we can factor it out by writing it on the outside of the parentheses and writing b plus c inside the parentheses.

Now, let's look at factorization. Being able to factor a polynomial is useful for simplifying polynomials, canceling common factors, and can help identify information, such as intercepts, on graphs. The process of finding common factors of a polynomial works like finding common factors of numbers.

To "factor" means to break down a number into smaller numbers that, when multiplied together, give us the original number. So for example, 30 is equal to 2 times 3 times 5. 2, 3, and 5 multiplied together give us 30. 2, 3, and 5 are also the prime factorization of 30 because 2, 3, and 5 are all prime numbers. Writing the prime factorization of all terms in a polynomial helps identify common factors and the greatest common factor.

Now, let's do some examples factoring polynomials. We have the expression 6x to the third plus 3x plus 15. To factor this polynomial, we need to factor or break down each individual term. 6x to the third can be written as 2 times 3 times x times x times x. 2 times 3 are the prime factors of 6, and x to the third means x multiplied by itself three times.

3x is broken down to 3 times x. 15 can be broken down or factored as 3 times 5. Looking at these three factorizations, we see that the common factor of each of these terms is 3. If terms have two or more common factors, the factors can be multiplied together to find the greatest common factor.

Let's do some more examples. Here, we have the polynomial 10x to the third minus 4x squared plus 4x. To factor this polynomial, we need to factor or break down each of the three terms. 10x to the third can be broken down to be 2 times 5 times x times x times x. Negative 4x squared can be broken down as negative 2 times 2 times x times x, and 4x can be broken down as 2 times 2 times x.

Now, we see that each of our terms have two common factors. They each have a positive 2, and they each have an x. So to find our greatest common factor, we multiplied these two common factors, 2 and x. So our greatest common factor is 2x.

Once we have factored out the greatest common factor, the expression inside the parentheses will be the remaining factors of each term. So from our first term, our remaining factors are 5x and x or 5x squared. In our second term, the remaining factors are a negative 2 and x. So negative 2x. And in our last term, the remaining factor is 2.

We can check to see that we have factored the polynomial correctly by distributing the greatest common factor back into each term inside the parentheses. So 2x times 5x squared is 10x to the third. 2x times negative 2x is negative 4x squared, and 2x times 2 is a positive 4x. So because we have arrived back at our original expression, we have factored our polynomial correctly.

Here is our last example. We want to factor 12x squared minus 18x plus 6. To factor this polynomial, we're going to factor each of the three terms. 12x squared can be broken down to be 2 times 2 times 3 times x times x. Negative 18x can be broken down to be negative 2 times 3 times 3 times x, and 6 can be broken down to 2 times 3.

Here, again, we see that we have two common factors in each term. Each term has a 2 and a 3. Multiplying these common factors together gives us our greatest common factor of 6. So 6 is our greatest common factor, and we write it on the outside of the parentheses. And in the parentheses, we write the remaining factors of each term.

So in the first term, our remaining factors are 2x and x or 2x squared. In the second term, our remaining factors are negative 3 and an x. So negative 3x. In our last term, we see that we have factored out both of the factors of 6, which means that we only have a 1 remaining.

Again, we can check to see that we have factored our polynomial correctly by distributing the 6 back into each of the three terms in the parentheses. So 6 times 2x squared is 12x squared. 6 times negative 3x is negative 18x, and 6 times 1 is 6. So we see that we have indeed factored our polynomial correctly.

Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. A "polynomial" is an expression involving one or more terms.

Coefficients and variables of a term are factors of that term. Factoring is the reverse process of distributing where a common factor is factored out of two or more terms and written outside of the parentheses. If terms have two or more common factors, the factors can be multiplied together to find the greatest common factor.

So I hope these important points and examples helped you understand a little bit more about finding the greatest common factor of a polynomial. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.