Question
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
One factor of x squared plus x plus 30 is xplus5.
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Part 1
Choose the correct answer below.
A.
The statement is true.
B.
The statement is false. An actual factor is xminus5.
C.
The statement is false. An actual factor is xplus6.
D.
The statement is false. The polynomial cannot be factored.
One factor of x squared plus x plus 30 is xplus5.
Question content area bottom
Part 1
Choose the correct answer below.
A.
The statement is true.
B.
The statement is false. An actual factor is xminus5.
C.
The statement is false. An actual factor is xplus6.
D.
The statement is false. The polynomial cannot be factored.
Answers
GPT-4o mini
To determine whether the statement is true or false, we need to analyze the polynomial \(x^2 + x + 30\) and see if \(x + 5\) is a factor.
To do this, we can use polynomial division or check by substituting \(x = -5\) into the polynomial. If \(x + 5\) is a factor, then \(x = -5\) should yield a result of 0.
Let's substitute \(x = -5\):
\[
(-5)^2 + (-5) + 30 = 25 - 5 + 30 = 50
\]
Since \(50\) is not \(0\), \(x + 5\) is not a factor of \(x^2 + x + 30\).
Next, we'll try to see if the polynomial can be factored at all. The factors would be \( (x + p)(x + q) \) where \( p + q = 1 \) (the coefficient of \(x\)) and \( p \cdot q = 30 \).
We can look for integer pairs \((p, q)\) that multiply to 30 and add to 1. The pairs are:
- \(1 \cdot 30\)
- \(2 \cdot 15\)
- \(3 \cdot 10\)
- \(-1 \cdot -30\)
- \(-2 \cdot -15\)
- \(-3 \cdot -10\)
None of these pairs provides a sum of 1, indicating that \(x^2 + x + 30\) cannot be factored over the integers.
Therefore, the correct conclusion is:
**D. The statement is false. The polynomial cannot be factored.**
To do this, we can use polynomial division or check by substituting \(x = -5\) into the polynomial. If \(x + 5\) is a factor, then \(x = -5\) should yield a result of 0.
Let's substitute \(x = -5\):
\[
(-5)^2 + (-5) + 30 = 25 - 5 + 30 = 50
\]
Since \(50\) is not \(0\), \(x + 5\) is not a factor of \(x^2 + x + 30\).
Next, we'll try to see if the polynomial can be factored at all. The factors would be \( (x + p)(x + q) \) where \( p + q = 1 \) (the coefficient of \(x\)) and \( p \cdot q = 30 \).
We can look for integer pairs \((p, q)\) that multiply to 30 and add to 1. The pairs are:
- \(1 \cdot 30\)
- \(2 \cdot 15\)
- \(3 \cdot 10\)
- \(-1 \cdot -30\)
- \(-2 \cdot -15\)
- \(-3 \cdot -10\)
None of these pairs provides a sum of 1, indicating that \(x^2 + x + 30\) cannot be factored over the integers.
Therefore, the correct conclusion is:
**D. The statement is false. The polynomial cannot be factored.**