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13 days ago

Describe the steps you would use to factor 2x3 + 5x2 – 8x – 20 completely. Then state the factored form.

2 answers:

5 0

Given the polynomial 2x³ + 5x² - 8x - 20 = 0,
follow these steps to factor:
1. Group terms and factor out the Greatest Common Factor (GCF) from each group, which are x² and 4:
x²(2x + 5) - 4(2x + 5) = 0
2. Factor out the common binomial:
(x² - 4)(2x + 5) = 0
3. Factor (x² - 4) using the difference of squares:
(x - 2)(x + 2)(2x + 5) = 0
4. Solutions are:
x = 2
x = -2
x = -5/2
This process uses factoring by grouping and the difference of squares method.

PIT_PIT [3.9K]13 days ago

5 0

The polynomial $2x^3 + 5x^2-8x-20$ could have roots among the divisors of -20, which include $\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20$ .
Testing x=1 yields $2\cdot1^3 + 5\cdot1^2-8\cdot1-20=-20\neq 0$ .
Testing x=-1 yields $2\cdot(-1)^3 + 5\cdot(-1)^2-8\cdot(-1)-20=-9\neq 0$ .
Testing x=2 gives $2\cdot2^3 + 5\cdot2^2-8\cdot2-20=0$ , confirming x=2 is a root and providing the factor (x-2).
Testing x=-2 results in $2\cdot(-2)^3 + 5\cdot(-2)^2-8\cdot(-2)-20=0$ , showing x=-2 is a root as well and giving the factor (x+2).
With factors (x-2) and (x+2) identified for $2x^3 + 5x^2-8x-20$ , the polynomial $x^2-4$ divides $2x^3 + 5x^2-8x-20$ . Dividing $2x^3 + 5x^2-8x-20$ by $x^2-4$ produces:
$2x^3 + 5x^2-8x-20=(x-2)(x+2)(2x+5)$ .

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