Which polynomial function has a leading coefficient of 1, roots –2 and 7 with multiplicity 1, and root 5 with multiplicity 2?
a) f(x) = 2(x + 7)(x + 5)(x – 2)
b) f(x) = 2(x – 7)(x – 5)(x + 2)
c) f(x) = (x + 7)(x + 5)(x + 5)(x – 2)
d) f(x) = (x – 7)(x – 5)(x – 5)(x + 2)
Answers
Option a:
f(x) = 2(x + 7)(x + 5)(x – 2)
⇒ The leading coefficient is 2
⇒ Not the answer we are looking for
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Option b:
f(x) = 2(x – 7)(x – 5)(x + 2)
⇒ The leading coefficient is 2
⇒ Not the answer we are looking for
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Option c:
f(x) = (x + 7)(x + 5)(x + 5)(x – 2)
⇒ Leading coefficient = 1
Find the roots:
(x + 7)(x + 5)(x + 5)(x – 2) = 0
(x + 7)(x + 5)²(x – 2) = 0
x = -7 or x = -5 or x = 2
⇒ The roots are 2 and -7 with multiplicity 1, and root - 5 with multiplicity 2
⇒ Not the answer we are looking for
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Option d:
f(x) = (x – 7)(x – 5)(x – 5)(x + 2)
⇒ Leading coefficient = 1
Find the roots:
(x – 7)(x – 5)(x – 5)(x + 2) = 0
(x – 7)(x – 5)²(x + 2) = 0
x = 7 or x = 5 or x = -2
⇒ The roots are -2 and 7 with multiplicity 1, and root 5 with multiplicity 2
⇒ This is the answer we are looking for
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Answer: (d) f(x) = (x – 7)(x – 5)(x – 5)(x + 2)
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