Factor the following polynomials.
9. 3x2 – 12
10. 5x3y + 35x2y + 45xy
To factor a polynomial, we want to express it as a product of two or more simpler polynomials.
Let's start with the first polynomial:
9. 3x^2 - 12
First, we look for a common factor that can be factored out from both terms. In this case, the common factor is 3:
3(x^2 - 4)
Now, we can further factor the expression inside the parentheses. It is a difference of squares, which can be factored as:
3(x - 2)(x + 2)
So the factored form of the polynomial 3x^2 - 12 is 3(x - 2)(x + 2).
Moving on to the second polynomial:
10. 5x^3y + 35x^2y + 45xy
First, we look for a common factor that can be factored out from all the terms. In this case, the common factor is 5xy:
5xy(x^2 + 7x + 9)
Now, we can further factor the expression inside the parentheses. It is a quadratic trinomial that cannot be easily factored with integers. In this case, we can use the quadratic formula or complete the square to find the factors. However, since the question does not specify the need to fully factor the quadratic, we will leave it as is.
Therefore, the factored form of the polynomial 5x^3y + 35x^2y + 45xy is 5xy(x^2 + 7x + 9).