Omg ! given f(x)=x^4+4x^3-7x^2-22x+24. explain how to find the x & y-intercept and end behavior without graphing technology. also, explain how to find the x & y-intercept and end behavior with graphing technology

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.