Answers:
f(x) rises on left and right.
y-intercept = (0, 24)
x-intercepts: x = -4, -3, 1, 2
Explanation:
f(x) = x⁴ + 4x³ -7x² -22x + 24
This is a 4th degree polynomial in which the leading term is x⁴.
End behaviour
If x is large enough, the leading term will outweigh all the others.
If x is large and positive, x⁴ will be large and positive.
If x is large and negative, x⁴ will be large and positive.
Thus, f(x) rises to the left and to the right.
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y-Intercept
The y-intercept is the value of y when x = 0.
f(0) = 24
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x-Intercepts
f(x) is a 4th degree polynomial, so there can be no more than four real roots (zeros) .
The roots are the values of x that make f(x) = 0.
We can use the rational zeros theorem to find the zeroes.
a₀ = 24; a₄ = 1
The roots must be the factors of 24/1.
Factors of 24 = ±1, ±2, ±3, ±4, ±6, ±8, ±12
Factors of 1 = ±1
p/q = ±1, ±2, ±3, ±4, ±6, ±8, ±12
We use either long division or synthetic division to find the factors.
It’s a matter of trial and error with many possibilities, so I will give just those that work.
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Try x = 1.
Factor out (x-1)
(x⁴ + 4x³ -7x² -22x + 24)/(x-1) = x³ + 5x² - 2x – 24
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Try x = 2
Factor out (x-2)
(x³ + 5x² - 2x – 24)/(x-2) = x² + 7x + 12
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Factor the quadratic
x² + 7x + 12 = (x + 3)(x + 4)
f(x) = (x-1)(x-2)(x+3)(x+4)
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Set each root separately to zero and solve for x.
x = 1, 2, -3, -4
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WITH GRAPHING TECHNOLOGY
We can use graphing technology to get the graph below.
End behaviour
The graph rises on the left and right .
y-Intercept
The graph crosses the y-axis at y =24.
x-intercepts
The graph crosses the x-axis at -4, -3, 1, and 2.