Question
Which of the below binomials is a factor of the polynomial shown?
2b²+18b+40
*
(b+2)
(b+5)
(b+8)
(b+10)
2b²+18b+40
*
(b+2)
(b+5)
(b+8)
(b+10)
Answers
GPT-4o mini
To determine which binomial is a factor of the polynomial \( 2b^2 + 18b + 40 \), we can use the factored form of the polynomial. First, we can factor out the greatest common factor (GCF) from the polynomial.
The GCF of the coefficients \(2\), \(18\), and \(40\) is \(2\). We factor this out:
\[
2b^2 + 18b + 40 = 2(b^2 + 9b + 20)
\]
Now, we need to factor the quadratic \(b^2 + 9b + 20\) further.
To factor \(b^2 + 9b + 20\), we need to find two numbers that multiply to \(20\) (the constant term) and add up to \(9\) (the coefficient of \(b\)).
The pairs of factors of \(20\) are:
- \(1 \cdot 20\) (sums to \(21\))
- \(2 \cdot 10\) (sums to \(12\))
- \(4 \cdot 5\) (sums to \(9\))
The pair \(4\) and \(5\) meet our criteria:
\[
b^2 + 9b + 20 = (b + 4)(b + 5)
\]
So, the complete factorization of the polynomial is:
\[
2b^2 + 18b + 40 = 2(b + 4)(b + 5)
\]
The factors of \(2b^2 + 18b + 40\) are \(2\), \((b + 4)\), and \((b + 5)\).
Now we can see that the binomials \( (b + 4) \) and \( (b + 5) \) are the possible factors. The binomial \( (b + 5) \) is listed in your options. Therefore, the answer is:
**(b + 5)**
The GCF of the coefficients \(2\), \(18\), and \(40\) is \(2\). We factor this out:
\[
2b^2 + 18b + 40 = 2(b^2 + 9b + 20)
\]
Now, we need to factor the quadratic \(b^2 + 9b + 20\) further.
To factor \(b^2 + 9b + 20\), we need to find two numbers that multiply to \(20\) (the constant term) and add up to \(9\) (the coefficient of \(b\)).
The pairs of factors of \(20\) are:
- \(1 \cdot 20\) (sums to \(21\))
- \(2 \cdot 10\) (sums to \(12\))
- \(4 \cdot 5\) (sums to \(9\))
The pair \(4\) and \(5\) meet our criteria:
\[
b^2 + 9b + 20 = (b + 4)(b + 5)
\]
So, the complete factorization of the polynomial is:
\[
2b^2 + 18b + 40 = 2(b + 4)(b + 5)
\]
The factors of \(2b^2 + 18b + 40\) are \(2\), \((b + 4)\), and \((b + 5)\).
Now we can see that the binomials \( (b + 4) \) and \( (b + 5) \) are the possible factors. The binomial \( (b + 5) \) is listed in your options. Therefore, the answer is:
**(b + 5)**