Final answer:
The cubic polynomial h(x)=x³ - 7x² + 10x - 4 can be expressed as a product of its linear factors by factoring out (x-1) to obtain (x - 1)(x² - 6x + 4), and then using the quadratic formula to find the remaining factors resulting in (x - 1)(x - (3 + √5))(x - (3 - √5)).
Step-by-step explanation:
To express the polynomial h(x)=x³ - 7x² + 10x - 4 as a product of linear factors, knowing that 1 is a zero of h(x), we first factor out (x - 1) from h(x) using either polynomial division or synthetic division. However, the polynomial provided seems to have an error in the initial question since it's described as both cubic and quadratic in different parts. Assuming that the correct polynomial is actually cubic and that the initial coefficient is indeed 1 (as in x³), we start our process.
Let's begin with polynomial division to find a quotient polynomial q(x):
x² - 6x + 4
1 | x³ - 7x² + 10x - 4
- (x³ - x²)
- 6x² + 10x
- (-6x² + 6x)
4x - 4
-(4x - 4)
0
This gives us h(x) = (x - 1)(x² - 6x + 4). Now, we use the quadratic formula to solve for the other zeros from x² - 6x + 4. The quadratic formula is x = (-b ± √(b² - 4ac))/2a, where a=1, b=-6, and c=4 in our case.
Substituting the values we get:
x = (6 ± √(36 - 16))/2 = (6 ± √20)/2 = (6 ± 2√5)/2 = 3 ± √5
Therefore, the polynomial h(x) can be expressed as (x - 1)(x - (3 + √5))(x - (3 - √5)).