Hopelessly loss in New Joisey If Anyone can help me I greatly appericate it. Anyone know how do we find the greatest common factor of a polynomial? Can you give me an example,
showing your work.
2.When finding the greatest common factor of a polynomial, can it ever be larger than the smallest
coefficient?
3. Can it ever be smaller than the smallest coefficient?
4. Can anyone Provide me an example.
12x^3y^2 + 48x^6y^4 - 60xy^2
first look at the constant coefficients:
12, 48, 60, their hightest common factor is 12
now look at the x's
x^3 , x^6 , x , their highest common factor is x
and for the y's
y^2 , y^4, and y^2, the highest common factor is y^2
so
= 12xy^2(x^2 + 4x^5y^2 - 5)
always mentally expand your answer to make sure you get back the original.
Had my last term be -50xy^2
then the HCF of the coefficients would have to be only 2
With that you should be able to answer the remaining questions.
1. To find the greatest common factor of a polynomial, you can follow these steps:
a. Write down the polynomial in factored form.
b. Identify the common factors among all the terms of the polynomial.
c. Choose the factors that have the lowest exponent for each common factor.
d. Multiply the chosen factors together to find the greatest common factor.
Let's take an example to understand this better:
Example: Find the greatest common factor of the polynomial 6x^2 + 9x^3 - 12x^4.
Step 1: Write down the polynomial in factored form, if possible.
The given polynomial cannot be factored further, so it is already in factored form.
Step 2: Identify the common factors among all the terms.
In this case, there is no common factor among all the terms.
Step 3: Choose the factors with the lowest exponent for each common factor.
Since there are no common factors, we move on to the next step.
Step 4: Multiply the chosen factors together to find the greatest common factor.
Since there are no common factors, the greatest common factor of this polynomial is 1.
2. The greatest common factor of a polynomial cannot be larger than the smallest coefficient. This is because the greatest common factor is a factor that divides evenly into all the terms of the polynomial. Therefore, it cannot be larger than any of the coefficients in the polynomial.
3. Similarly, the greatest common factor cannot be smaller than the smallest coefficient. It must be a factor that divides evenly into all the terms, so it must be at least as large as the smallest coefficient.
4. Let's consider another example to understand this:
Example: Find the greatest common factor of the polynomial 2x^3 + 4x^2 - 6x.
Step 1: Write down the polynomial in factored form, if possible.
The given polynomial cannot be factored further, so it is already in factored form.
Step 2: Identify the common factors among all the terms.
The common factor among all the terms is 2x.
Step 3: Choose the factors with the lowest exponent for each common factor.
Since there is only one common factor, we skip this step.
Step 4: Multiply the chosen factors together to find the greatest common factor.
The greatest common factor of this polynomial is 2x.
So, the answer to this example is 2x.