Question

use technology to find points and then graph the following polynomial, then write it in factore form.
$f(x) = x^4 + 4x^3 - 28x^2 - 64x + 192$
plot all of the points that fit on the axes.
you may click a plotted point to delete it.

Explanation:

Step1: Find rational roots (Rational Root Theorem)

Possible rational roots are factors of 192 over factors of 1: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±32, ±48, ±64, ±96, ±192.
Test \( x = 2 \): \( f(2)=16 + 32 - 112 - 128 + 192 = 0 \). So \( (x - 2) \) is a factor.
Test \( x = -2 \): \( f(-2)=16 - 32 - 112 + 128 + 192 = 192
eq 0 \).
Test \( x = 3 \): \( f(3)=81 + 108 - 252 - 192 + 192 = -63
eq 0 \).
Test \( x = 4 \): \( f(4)=256 + 256 - 448 - 256 + 192 = 0 \). So \( (x - 4) \) is a factor.
Test \( x = -6 \): \( f(-6)=1296 - 864 - 1008 + 384 + 192 = 0 \). So \( (x + 6) \) is a factor.
Test \( x = -8 \): \( f(-8)=4096 - 2048 - 1792 + 512 + 192 = 960
eq 0 \).

Step2: Perform polynomial division or use synthetic division

Since \( x = 2, 4, -6 \) are roots, we can factor \( f(x) \) as \( (x - 2)(x - 4)(x + 6)(x + a) \). Expand \( (x - 2)(x - 4)(x + 6) = (x^2 - 6x + 8)(x + 6) = x^3 + 6x^2 - 6x^2 - 36x + 8x + 48 = x^3 - 28x + 48 \). Now divide \( f(x) \) by \( x^3 - 28x + 48 \):
\( x^4 + 4x^3 - 28x^2 - 64x + 192 \div (x^3 - 28x + 48) = x + 4 \). So the last factor is \( (x + 4) \).

Step3: Write in factored form

Thus, \( f(x)=(x - 2)(x - 4)(x + 4)(x + 6) \). We can also group as \( (x^2 - 4)(x^2 + 10x + 24)=(x - 2)(x + 2)(x + 4)(x + 6) \)? Wait, no, earlier division showed \( x + 4 \). Wait, recheck: when we factored with \( (x - 2)(x - 4)(x + 6) \), the quotient was \( x + 4 \), so correct factored form is \( (x - 2)(x - 4)(x + 4)(x + 6) \) or \( (x - 2)(x + 2)(x + 4)(x + 6) \)? Wait, \( (x - 2)(x + 2)=x^2 - 4 \), \( (x + 4)(x + 6)=x^2 + 10x + 24 \). Multiply \( (x^2 - 4)(x^2 + 10x + 24)=x^4 + 10x^3 + 24x^2 - 4x^2 - 40x - 96=x^4 + 10x^3 + 20x^2 - 40x - 96 \), which is not \( f(x) \). So earlier synthetic division: let's do synthetic for \( x = 2 \):
Coefficients: 1 | 4 | -28 | -64 | 192
Bring down 1. Multiply by 2: 2. Add to 4: 6. Multiply by 2: 12. Add to -28: -16. Multiply by 2: -32. Add to -64: -96. Multiply by 2: -192. Add to 192: 0. So quotient is \( x^3 + 6x^2 - 16x - 96 \). Now factor \( x^3 + 6x^2 - 16x - 96 \). Factor by grouping: \( x^2(x + 6) - 16(x + 6)=(x^2 - 16)(x + 6)=(x - 4)(x + 4)(x + 6) \). Ah, there we go! So \( f(x)=(x - 2)(x - 4)(x + 4)(x + 6) \).

Answer:

\( f(x) = (x - 2)(x - 4)(x + 4)(x + 6) \) (or \( (x - 2)(x + 2)(x + 4)(x + 6) \) is incorrect, correct is \( (x - 2)(x - 4)(x + 4)(x + 6) \) after proper factoring)