Final Answer:
p(x) = (x - 5)(x + 2)(x + 1)
By leveraging the known factor (x - 5) , the polynomial p(x) = 23x² - 210x - 20 can be expressed as a product of linear factors: (x - 5)(x + 2)(x + 1).
Step-by-step explanation:
In order to express the polynomial p(x) = 23x² - 210x - 20 as a product of linear factors, we leverage the information that (x - 5) is a known factor. The factorization process involves setting x = 5 to satisfy the factor theorem, which states that if (x - c) is a factor, then p(c) = 0 . In this case, p(5) = 0 , confirming (x - 5) as a factor.
Once (x - 5) is established, we can use polynomial division or other factoring methods to further break down the quadratic expression 23x² - 210x - 20 . The result is (x - 5)(x + 2)(x + 1) , representing the polynomial as a product of linear factors.
Understanding factorization techniques and the factor theorem is essential in efficiently decomposing polynomials and revealing their underlying linear factors.