Final answer:
The roots for the given polynomial equations are: x = 1, x = -3 and -3, x = -5 and 5, and x = 5/3.
Step-by-step explanation:
1. The first polynomial equation, 2x - 3x + 4 - 3 = 0, can be simplified to
-x + 1 = 0.
To find the root, we add x to both sides, resulting in
-x + x + 1 = 0 + x.
This simplifies to 1 = x.
So, the root of this equation is x = 1.
2. The second polynomial equation, x² + 6x + 9 = 0, is already in standard form.
We can factor this equation into
(x + 3)(x + 3) = 0.
To find the roots, we set each factor equal to zero:
x + 3 = 0 or x + 3 = 0. Solving for x gives us x = -3.
Therefore, the roots of this equation are x = -3 and x = -3.
3. The third polynomial equation, x² - 25 = 0, is a difference of squares.
We can factor this equation into
(x + 5)(x - 5) = 0.
Setting each factor equal to zero gives us
x + 5 = 0 or x - 5 = 0. Solving for x gives us x = -5 or x = 5.
Therefore, the roots of this equation are x = -5 and x = 5.
4. The fourth polynomial equation, x² + 8x + 15 = 0, can be factored into
(x + 3)(x + 5) = 0.
Setting each factor equal to zero gives us
x + 3 = 0 or x + 5 = 0.
Solving for x gives us x = -3 or x = -5.
Therefore, the roots of this equation are x = -3 and x = -5.
5. The fifth polynomial equation, 3x - 5 = 0, can be solved by isolating x.
Adding 5 to both sides of the equation gives us
3x - 5 + 5 = 0 + 5, which simplifies to 3x = 5.
Next, we divide both sides by 3 to solve for x:
3x/3 = 5/3.
The final equation is x = 5/3.
Hence, the root of this equation is x = 5/3.