1. How do we find the greatest common factor of a polynomial? Demonstrate the process with 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. Provide an example for the class to factor
1. To find the greatest common factor (GCF) of a polynomial, we need to identify the largest factor that is common to all the terms in the polynomial. Here's the step-by-step process:
Example: Let's find the GCF of the polynomial 4x^3 - 8x^2 + 12x.
Step 1: Write down the factors of each coefficient:
- Factors of 4: 1, 2, 4
- Factors of -8: 1, 2, 4, 8
- Factors of 12: 1, 2, 3, 4, 6, 12
Step 2: Identify the common factors:
The only common factor among these coefficients is 1.
Step 3: Write the GCF using the common factors:
GCF = 1
So, the GCF of the polynomial 4x^3 - 8x^2 + 12x is 1.
2. No, the greatest common factor (GCF) of a polynomial can never be larger than the smallest coefficient. This is because the GCF represents the largest factor that divides evenly into all the terms of the polynomial. If a factor is larger than the smallest coefficient, it would not be a common factor.
3. Yes, the GCF of a polynomial can be smaller than the smallest coefficient. The GCF only represents the largest factor that is common to all the terms, it doesn't necessarily have to be larger than any specific coefficient. It depends on the specific terms in the polynomial.
For example, if we have the polynomial 2x^2 - 4x, the GCF would be 2, which is smaller than both coefficients 2 and 4.
4. Sure! Here's an example for the class to factor:
Example: Factor the polynomial 6x^2 + 11x - 10.
To factor this polynomial, we need to find two binomials that, when multiplied, give us the original polynomial.
Step 1: Multiply the coefficient of the leading term (6) with the constant term (-10):
6 * (-10) = -60.
Step 2: Find two numbers that add up to the coefficient of the middle term (11) and multiply to -60:
Numbers: 15 and -4.
Step 3: Use the factored numbers to rewrite the middle term:
11x = 15x - 4x.
Step 4: Group the polynomial and factor by grouping:
6x^2 + 15x - 4x - 10.
(6x^2 + 15x) + (-4x - 10).
3x(2x + 5) - 2(2x + 5).
Step 5: Factor out the common binomial (2x + 5):
(2x + 5)(3x - 2).
So, the factored form of the polynomial 6x^2 + 11x - 10 is (2x + 5)(3x - 2).