Answer:
a = 1 and b = -12
Step-by-step explanation:
* Lets look to the graph to solve the problem
- f(x) is a rational function with:
# numerator ⇒ x² - 4x + 3
# denominator ⇒ x² + ax + b
- To find the values of a and b lets use the hole and the vertical
asymptote in the graph of the function
∵ There is a hole at x = 3
- The hole appears when the numerator and the denominator have
a common factor
∵ x = 3 ⇒ subtract 3 from both sides
∴ x - 3 = 0
∴ The numerator and the denominator have a factor (x - 3)
- The vertical asymptotes appear at the zeros of the denominator
that means the values of x when the denominator = 0
∵ There is a vertical asymptote at x = -4
∵ x = -4 ⇒ add 4 for both sides
∴ x + 4 = 0
∴ The denominator has another factor (x + 4)
- The denominator x² + ax + b has two factors (x - 3) and (x + 4) lets
multiply them to find the value of a and b
∵ (x - 3)(x + 4) = (x)(x) + (x)(4) + (-3)(x) +(-3)(4)
∴ (x - 3)(x + 4) = x² + 4x - 3x - 12 = x² + x - 12
∴ x² + ax + b = x² + x - 12
- By comparing the two sides
∴ a = 1 and b = -12
