1. The expression log 5x is equivalent to
2. Write the equation of a polynomial whose sum of the roots equals -3 and the product of the roots equals 9\2?
P.S. - I have been trying to solve these problems on my own and have not been successful at it. Can someone please help ASAP? Thanks.
1. log5 + logx
2. See
http://www.mathsisfun.com/algebra/polynomials-sums-products-roots.html
1. To simplify the expression log 5x, we can use the logarithmic property that states log a(b) = log a + log b. Applying this property, we can rewrite the expression as log 5 + log x.
2. To write the equation of a polynomial given the sum and product of its roots, we can use Viète's formulas. Let's consider a polynomial of degree n:
p(x) = a₀ + a₁x + a₂x² + ... + aₙ₋₁xⁿ⁻¹ + aₙxⁿ
The sum of the roots, denoted as S, is given by:
S = -a₁/a₀
The product of the roots, denoted as P, is given by:
P = (-1)ⁿ aₙ/a₀
For our specific problem, where the sum of the roots equals -3 and the product of the roots equals 9\2, we have:
S = -3
P = 9\2
Since the sum of the roots is -3, we can write this as:
-a₁/a₀ = -3
From this equation, we can solve for a₁ in terms of a₀:
a₁ = 3a₀
Similarly, since the product of the roots is 9\2, we have:
(-1)ⁿ aₙ/a₀ = 9\2
Simplifying:
(-1)ⁿ aₙ = 9\2
Now, we have two equations:
a₁ = 3a₀
(-1)ⁿ aₙ = 9\2
These equations allow us to express all the coefficients of the polynomial in terms of a₀. Therefore, we can write the equation of the polynomial as:
p(x) = a₀ + 3a₀x + ... + [(-1)ⁿ aₙ]xⁿ
To determine the values of a₀, a₁, and aₙ, we need additional information about the polynomial, such as the degree or specific coefficients.