Final answer:
The polynomials are factorized using the splitting the middle term method to find factors that satisfy the conditions for products and sums. Once factorized, the zeros are determined by setting each factor equal to zero. Each polynomial yields two zeros which represent where the graph would cross the x-axis.
Step-by-step explanation:
Factoring polynomials and finding their zeros involves identifying the values that satisfy the equation when the polynomial is equal to zero. The splitting the middle term method is one such technique used to factorize quadratic polynomials. Below, we use this method to factorize each given polynomial and find their zeros.
Factorization and Zeros of Polynomials:
- p(y) = y² - 3y + 2: To factorize, we need two numbers whose product is +2 and whose sum is -3. The numbers -2 and -1 satisfy this condition. Thus, p(y) can be written as (y - 2)(y - 1). The zeros are y = 2 and y = 1.
- p(x) = 5x² + 9x + 4: Here, we look for two numbers whose product is 5*4=20 and whose sum is 9. The numbers 5 and 4 fulfill this criterion. We can split the 9x into 5x + 4x and factor by grouping, resulting in (5x + 4)(x + 1). The zeros are x = -4/5 and x = -1.
- p(x) = x² + 4x - 12: We need two numbers with a product of -12 and a sum of 4. The numbers 6 and -2 work. So p(x) becomes (x + 6)(x - 2), and the zeros are x = -6 and x = 2.
For each polynomial, we found the necessary factors and then the zeros of the polynomial, which are the points where the graph of the polynomial would intersect the x-axis on a coordinate plane.
To ensure our factorization and zeros are correct, we can check the answer by plugging in the zeros into the original equation and ensuring it equals zero.