Respuesta :
This is the remainder factor theorem.
The theorem (I don't remember exactly) basically says something about how if you divide P(x) by (x+b), the remainder can be found by P(-b).
So in this case, since its p(3), this means that -b = 3
b = -3
So x+b = x-3.
That means p(x) is divided by x-3. There is a remainder of -2 when P(x) is divided by this. Therefore the last one is correct.
The rest cannot be correct as we have no information about them
The theorem (I don't remember exactly) basically says something about how if you divide P(x) by (x+b), the remainder can be found by P(-b).
So in this case, since its p(3), this means that -b = 3
b = -3
So x+b = x-3.
That means p(x) is divided by x-3. There is a remainder of -2 when P(x) is divided by this. Therefore the last one is correct.
The rest cannot be correct as we have no information about them
The remainder when P(x) is divided by (x − 3) is −2.
Further explanation
Given:
For a polynomial P(x), the value of P(3) is −2.
Question:
Which of the following statements must be true about P(x)?
Problem-solving:
We will solve the problem of the remainder theorem.
Consider the following division:
[tex]\boxed{ \ P(x) = (x - a) \cdot Q(x) + R(x) \ }[/tex]
- P(x) is called the dividend.
- (x - a) or D(x) is called the divisor.
- Q(x) is called the quotient.
- R(x) is called the remainder.
The remainder theorem says that:
when P(x) is divided by (x - a), the remainder is P(a).
Let R(x) = r, a constant polynomial. Substituting a for x gives us
[tex]\boxed{ \ P(k) = (a - a) \cdot Q(a) + r \ }[/tex]
[tex]\boxed{ \ P(a) = r \ }[/tex]
From our case, the value of P(3) is −2.
Meaning:
- a = 3
- The divisor is [tex]\boxed{ \ (x - a) \rightarrow (x - 3) \ }[/tex]
- The remainder is [tex]\boxed{ \ r = -2 \ }[/tex]
Hence this can be concluded that the remainder when P(x) is divided by x − 3 is −2.
- - - - - - - - - -
A note on the factor theorem:
Let P(x) be polynomial. Then the following statements are true.
- If P(x) has a factor (x - a) then P(a) = 0.
- If P(a) = 0 then (x - a) is a factor of P(x).
From these options:
A). x − 5 represents a factor of P(x),
B). x − 2 represents a factor of P(x), and
C). x + 2 represents a factor of P(x),
we can say that all three options are related to the factor theorem.
A). x − 5 is a factor of P(x), therefore [tex]\boxed{ \ P(5) = 0 \ }[/tex]
B). x − 2 is a factor of P(x), therefore [tex]\boxed{ \ P(2) = 0 \ }[/tex]
C). x + 2 is a factor of P(x), therefore [tex]\boxed{ \ P(- 2) = 0 \ }[/tex]
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Keywords: polynomial P(x), the value of P(3), the dividend, the divisor, the quotient, the remainder theorem, which of the following must be true, a factor.
