Answer:
The x-intercepts are x = 1 , x = 2 , x = 3
The y-intercept is -6
Explanation:
* Lets explain how to solve the problem
- To find the x-intercept of a function substitute f(x) by 0
- To find the y-intercept of a function substitute x by 0
- To find the factors of quadratic function use the long division
* Lets solve the problem
∵ f(x) = x³ - 6x² + 11x - 6
∵ (x - 3) is one of its factors
- Use the long division to find the other factors
∵ x³ - 6x² + 11x - 6 ⇒ dividend
∵ x - 3 ⇒ divisor
# Divide the 1st term in the dividend by the 1st term of the divisor
∵ x³ ÷ x = x²
# Multiply x² by the divisor (x - 3)
∵ x²(x - 3) = x³ - 3x²
# subtract it from the dividend
∵ (x³ - 6x² + 11x - 6) - (x³ - 3x²) = (x³ - x³) + (-6x² + 3x²) +11x - 6
∴ The dividend is -3x² + 11x - 6
# Divide the 1st term in the dividend by the 1st term of the divisor
∵ -3x² ÷ x = -3x
# Multiply -3x by the divisor (x - 3)
∴ -3x(x - 3) = -3x² + 9x
# subtract it from the dividend
∵ (-3x² + 11x - 6) - (-3x² + 9x) = (-3x² - 3x²) + (11x - 9x) - 6
∴ The dividend is 2x - 6
# Divide the 1st term in the dividend by the 1st term of the divisor
∵ 2x ÷ x = 2
# Multiply 2 by the divisor (x - 3)
∴ 2(x - 3) = 2x - 6
# subtract it from the dividend
∴ (2x - 6) - (2x - 6) = (2x - 2x) + (-6 + 6) = 0
∴ (x³ - 6x² + 11x - 6) ÷ (x - 3) = x² - 3x + 2
∴ The factors of f(x) are (x - 3)( x² - 3x + 2)
- The factor (x² - 3x + 2) can factorize into two bracket
∵ The last term is positive and the middle term is negative than the
two brackets have the middle sign (-)
∵ x × x = x² ⇒ 1st terms in the two brackets
∵ 2 × 1 = 2 ⇒ 2nd terms in the two brackets
∵ 2 × x = 2x
∵ 1 × x = x
∵ 2x + x = 3x
∴ (x² - 3x + 2) = (x - 2)(x - 1)
∴ f(x) = (x - 3)(x - 2)(x - 1)
- To find the x-intercept put f(x) = 0
∴ (x - 3)(x - 2)(x - 1) = 0
- That means each bracket = 0
∵ x - 3 = 0 ⇒ add 3 for both sides
∴ x = 3
∵ x - 2 = 0 ⇒ add 2 for both sides
∴ x = 2
∵ x - 1 = 0 ⇒ add 1 for both sides
∴ x = 1
∴ The x-intercepts are x = 1 , x = 2 , x = 3
- To find the y-intercept put x = 0
∵ f(x) = x³ - 6x² + 11x - 6
∵ x = 0
∴ f(0) = 0 - 0 + 0 - 6 = -6
∴ The y-intercept is -6