Question

Consider the polynomials: f(x) = (3x – 5)(5x – 4) + (5 – 3x)(x + 2), g(x) = 9x^2 – 30x + 25.

Factorize f(x) and g(x).
Solve each of these equations: a) f(x) = 0; b) f(x) = g(x).
Consider the fractional expression: h(x) = f(x) / [8(x)].
a) For what values of the real number x is h(x) defined?
b) Simplify h(x).
c) Solve the equation h(x) = 0.

Answer 1

To factorize f(x) and g(x), set f(x) = 0 and f(x) = g(x), determine the values for which h(x) is defined, simplify h(x), and solve the equation h(x) = 0.

Step-by-step explanation:

To factorize the polynomial f(x) = (3x – 5)(5x – 4) + (5 – 3x)(x + 2), we can simplify the expression inside the brackets and combine like terms:

(15x2 – 19x + 20) + (5 – 3x)(x + 2)

Now, to solve the equations f(x) = 0 and f(x) = g(x), we can set the expressions equal to zero and solve for x:

(15x2 – 19x + 20) + (5 – 3x)(x + 2) = 0

9x2 – 30x + 25 = 0

To determine the values for which h(x) is defined, we need to find the values that make the denominator equal to zero. In this case, the denominator is 8x, so h(x) is defined for all values of x except x = 0. To simplify the expression h(x), we can divide f(x) by 8x:

h(x) = (15x2 – 19x + 20) + (5 – 3x)(x + 2) / [8(x)]

To solve the equation h(x) = 0, we can set the expression equal to zero and solve for x:

(15x2 – 19x + 20) + (5 – 3x)(x + 2) / [8(x)] = 0