For a polynomial p(x), the value of p(3) is −2. Which of the following must be true about p(x)?
A) x−5 is a factor of p(x).
B) x−2 is a factor of p(x).
C) x+2 is a factor of p(x).
D) The remainder when p(x) is divided by x−3 is −2.
Answers
The true statement about p(x) is A.
Given: For a polynomial p(x), the value of p(3) is −2.
To find: We have to find p(x).
Solution:
A) If we put x=3 then we get,
3-5=-2.
So, this is the function of p(x) and true about p(x).
B) If we put x=3 then we get,
3-2=1.
So, this is not the function of p(x) and is not true about p(x).
C) If we put x=3 then we get,
3+2=5.
So, this is not the function of p(x) and is not true about p(x).
D) If we put x=3 then we get,
3-3=0.
So, this is not the function of p(x) and is not true about p(x).
The remainder when p(x) is divided by x−3 is −2 if p(3) = - 2.
Given:
- A polynomial p(x)
- p(3) = - 2
To Find:
- Which of the following must be true about p(x)
- A) x−5 is a factor of p(x).
- B) x−2 is a factor of p(x).
- C) x+2 is a factor of p(x).
- D) The remainder when p(x) is divided by x−3 is −2.
Solution:
- Remainder Theorem. polynomial p(x) divided by x -a then ,
- p(a) = remainder
- Also, if x – a is a factor of p(x), then p(a) = 0,
- where a is any real number.
Step 1:
p(3) = - 2
=> if p(x) divided by x -3 then remainder = - 2
Hence The remainder when p(x) is divided by x−3 is −2 Must be True
option D is correct
Step 2:
Taking example p(x) = x² - 11
p(3) = 3² - 11 = - 2
Verification using long division
x + 3
x - 3 ) x² - 11 (
x² - 3x
______
3x - 11
3x - 9
______
-2
Remainder = - 2 verified
p(x) = x² - 11 , Does not have x−5 is a factor of p(x)
Hence option A) not true
p(x) = x² - 11 , Does not have x−2 is a factor of p(x)
Hence option B) not true
p(x) = x² - 11 , Does not have x+2 is a factor of p(x)
Hence option C) not true
So only option which must be True is option D)
The remainder when p(x) is divided by x−3 is −2.