To find the factors of the quadratic polynomial \( 3x^2 + 23x - 8 \), we are looking for two binomials of the form \( (ax + b)(cx + d) \) such that when multiplied out, they give us the original polynomial. Specifically, we want to find values of \( a, b, c, \) and \( d \) such that:
1. \( ac = 3 \), because this will give us the coefficient of \( x^2 \) term.
2. \( bd = -8 \), because this will give us the constant term.
3. \( ad + bc = 23 \), because this will give us the middle term, which is the coefficient of the \( x \) term.
Since we want \( ac = 3 \) and 3 is a prime number, the only factors are 1 and 3 (since we're dealing with integer coefficients). Therefore, \( a \) and \( c \) can only be 3 and 1, in some order, without loss of generality, we can take \( a = 3 \) and \( c = 1 \).
We now need to find the factors of \( bd = -8 \). The pairs of factors of 8 are (1, 8), (2, 4) considering the signs, since one factor needs to be negative to make the product -8, we have (-1, 8), (1, -8), (-2, 4), and (2, -4).
Finally, we want the combination where \( ad + bc = 23 \). Testing the factors we've listed:
- With \( (3x - 1)(x + 8) \), we get \( ad + bc = -3 + 8 = 5 \), which is not correct.
- With \( (3x + 1)(x - 8) \), we get \( ad + bc = 3 - 8 = -5 \), which is also not correct.
- With \( (3x - 2)(x + 4) \), we get \( ad + bc = -6 + 4 = -2 \), which is incorrect as well.
The last pair to check is \( (3x + 4)(x - 2) \), and for this pair:
\( ad + bc = -6 + 12 = 6 \), which is not correct.
Since none of the combinations matched, it seems we must re-evaluate our choice of signs for factors of -8.
Testing other possible combinations with different placement of negative signs:
- With \( (3x - 4)(x + 2) \), we get \( ad + bc = -12 + 6 = -6 \), which is incorrect.
- Finally, we will test the factors \( (3x + 2)(x - 4) \). For this pair, we get \( ad + bc = (-4)(3) + (2)(1) = -12 + 2 = -10 \), which is also incorrect.
It appears that there was a calculation mistake in the previous step. Let’s correct that. Let’s go through the pairings again.
- With \( (3x - 4)(x + 2) \), we get \( ad + bc = (3)(2) + (-4)(1) = 6 - 4 = 2 \), which is incorrect.
- With \( (3x - 2)(x + 4) \), we get \( ad + bc = (3)(4) + (-2)(1) = 12 - 2 = 10 \), which is again incorrect.
- With \( (3x + 4)(x - 2) \), we get \( ad + bc = (3)(-2) + (4)(1) = -6 + 4 = -2 \), which is incorrect.
The pairs of factors we have tried do not yield the correct coefficient for the \( x \) term. It seems like we made a computation error in evaluating the possible pairings. We have to find the correct combination of factors of -8 that also satisfy the equation \( 3d + b = 23 \). Let’s correct the pairings:
- With \( (3x + 8)(x - 1) \), we get \( ad + bc = (3)(-1) + (8)(1) = -3 + 8 = 5 \), which is incorrect.
- Now, consider \( (3x - 8)(x + 1) \), we get \( ad + bc = (3)(1) + (-8)(1) = 3 - 8 = -5 \), which gives the wrong sign, but we see we've made an error in the sign of the middle term.
We need to ensure the middle term has a positive coefficient.
In light of this, let's correct the pairing:
For \( (3x - 8)(x + 1) \), we get \( ad + bc = (3)(1) + (-8)(1) = 3 - 8 = -5 \), which is not correct.
It appears there's an error in the initial assumptions. When dealing with prime numbers like 3, the prime number itself and 1 are not the only factors we need to consider, since we can also have negative factors. The correct factors could be -3 and -1 if signs are important.
Recalculating with this in mind and considering the original factor pairs for -8 again, we should look for a pair that adds up to 23 when multiplied by 3 and 1, respectively, and still gives a product of -8. The correct pairing is thus \( (3x - 1)(x + 8) \), but since this did not give us the correct middle term, we must re-evaluate it.
Let's evaluate it one more time:
For \( (3x - 1)(x + 8) \), the middle term, when expanded, would be \( -1*x + 3*8x = -x + 24x = 23x \), which actually matches our requirement of a middle term of 23x. The constant term would be \( -1*8 = -8 \), and the \( x^2 \) term would be \( 3x^2 \), which matches our original polynomial.
Therefore, the correct factors of \( 3x^2 + 23x - 8 \) are indeed \( (3x - 1)(x + 8) \), which is not listed as one of the options you provided. It seems there was perhaps a mistake in the given options or in the transcribing of the polynomial.
