Factor each polynomial. Write prime if it is not factorable. Check your answer.
1.49m²-27m⁴+49m²-7
2.12c³d-15c²d+3cd
#1. Combining the m^2 terms, you have
98m^2 - 27m^4 - 7
If you let u = m^2, that's
-(27u^2-98u+7)
Since the discriminant is not a perfect square, it will not factor
#2. 3cd is clearly a common factor, so
3cd(4c^2 - 5c + 1)
You should be able to factor that.
Djjd
To factor each polynomial, let's take them one at a time:
1. The polynomial 49m² - 27m⁴ + 49m² - 7 can be simplified by combining like terms:
= (49m² + 49m²) - 27m⁴ - 7
= 98m² - 27m⁴ - 7
At first glance, it doesn't seem like this polynomial can be factored further using common methods such as factoring out a common binomial or trinomial. To make sure, let's check if it can be factored using the difference of squares.
Using the difference of squares, we can rewrite the polynomial as:
= (7m)² - (3m²)² - 7
However, this doesn't factor further using the difference of squares or any other common factoring methods. So the factored form of the polynomial is prime.
Prime.
2. The polynomial 12c³d - 15c²d + 3cd can also be simplified by combining like terms:
= 12c³d - (15c²d - 3cd)
Now, notice that we can factor out a common term "cd" from both terms:
= cd(12c² - 15c + 3)
Now we can focus on factoring the quadratic expression inside the parentheses:
= cd(3c - 3)(4c - 1)
Therefore, the factored form of the polynomial is: cd(3c - 3)(4c - 1).
cd(3c - 3)(4c - 1).
To factor each polynomial, we need to identify any common factors and then use factoring techniques if possible. Let's start with the first polynomial:
1. 49m² - 27m⁴ + 49m² - 7
First, we can notice that the two terms "49m²" can be combined. So, let's rewrite the polynomial:
= (49m² + 49m²) - 27m⁴ - 7
Now, we can try to factor out a common factor from each term. In this case, the common factor is 7:
= 7(7m² + 7m²) - 27m⁴ - 7
Next, we can factor out the common factor of 7m² from the first two terms inside the parentheses:
= 7(7m² + 7m²) - 27m⁴ - 7
= 7(2m²) - 27m⁴ - 7
= 14m² - 27m⁴ - 7
The polynomial does not appear to have any common factors that we can further factor out. Therefore, the factored form of the polynomial is:
49m² - 27m⁴ + 49m² - 7 = 14m² - 27m⁴ - 7
As for the second polynomial:
2. 12c³d - 15c²d + 3cd
We can notice that each term has a common factor of cd. Let's factor that out:
= cd(12c² - 15c + 3)
The remaining quadratic expression inside the parentheses can be factored further. It seems like we can factor out a common factor of 3:
= cd(3(4c² - 5c + 1))
Now, we can try to factor the quadratic expression inside the parentheses. It factors to:
= cd(3(4c - 1)(c - 1))
So, the factored form of the second polynomial is:
12c³d - 15c²d + 3cd = cd(3(4c - 1)(c - 1))
You can check your answer by multiplying the factors back together to see if you obtain the original polynomial.