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1 answer:
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Answer:
x = 1 or x = ± i
Step-by-step explanation:
Note the sum of the coefficients
1 - 1 + 1 - 1 = 0
This indicates that x = 1 is a root, thus (x - 1) is a factor
Using long division or synthetic division, then
x³ - x² + x - 1 = (x - 1)(x² + 1), thus
(x - 1)(x² + 1) = 0
Equate each factor to zero and solve for x
x - 1 = 0 ⇒ x = 1
x² + 1 = 0 ⇒ x² = - 1 ⇒ x = ± = ± i
Using the Factor Theorem, the polynomials are given as follows:
1.
2.
3. P(x) = -0.1(x³ - 4x² - 3x + 18)
<h3>What is the Factor Theorem?</h3>The Factor Theorem states that a polynomial function with roots is given by:
In which a is the leading coefficient.
Item a:
The parameters are:
Hence the equation is:
P(x) = (x - 1)²x²(x + 4)
P(x) = (x² - 2x + 1)(x + 4)x²
P(x) = (x³ + 2x² - 7x + 1)x²
Item b:
The roots are:
Hence:
P(x) = a(x - 4)²x(x + 4)
P(x) = a(x² - 16)x(x - 4)
P(x) = a(x³ - 16x)(x - 4)
It passes through the point x = 5, P(x) = 36, hence:
45a = 36.
a = 4/5
a = 0.8
Hence:
Item 3:
The roots are:
Hence:
P(x) = a(x - 3)²(x + 2)
P(x) = a(x² - 6x + 9)(x + 2)
P(x) = a(x³ - 4x² - 3x + 18)
For the y-intercept, x = 0, y = -1.8, hence:
18a = -1.8 -> a = -0.1
Thus the function is:
P(x) = -0.1(x³ - 4x² - 3x + 18)
More can be learned about the Factor Theorem at brainly.com/question/24380382
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