Answer:
C- Omar evaluates 4x⁴−12x²−18x−162 when x = 3. He determines that the value of the expression is 0 and concludes that x−3 is a factor.
Step-by-step explanation:
Factor Theorem can be defined as a method or theorem that is used in mathematics to simplify polynomials. It is used to confirm if a particular root is the real root of the polynomial.
Factor Theorem states that:
a) If p(x) is a polynomial of degree of n is greater than 1 (n > 1) and a is any real number, then
b) x – a is a factor of p(x)
if p(a) = 0, x – a is a factor of p(x).
In the question above, we are told that:
Omar uses the factor theorem to determine whether x−3 is a factor of 4x⁴ −12x² − 18x − 162.
Using Factor Theorem,
b) x – a is a factor of p(x)
if p(a) = 0, x – a is a factor of p(x).
x - 3 = 0, x = 3
p(x) = 4x⁴ −12x² − 18x − 162 = 0
p(3) = 4(3)⁴ - 12(3)² - 18(3) - 162 = 0
p(3) = 4(81) - 12(9) - 18(3) - 162 = 0
p(3) = 324 - 108 - 54 - 162 = 0
p(3) = 324 - 324 = 0
p(3) = 0 = 0
Hence, since p(3) = 0, x - 3 is a factor of polynomial 4x⁴ −12x² − 18x − 162 .
Therefore,Option C- "Omar evaluates 4x⁴−12x²−18x−162 when x = 3. He determines that the value of the expression is 0 and concludes that x−3 is a factor. " is the correct answer.
