Question

which of the following expressions is a factor of the polynomial below? $8c^2 + 10c - 7$ a) $(4c + 1)$ the museum b) $(c - 7)$ the airport

Explanation:

Step1: Factor the quadratic polynomial

We use the AC method to factor \(8c^{2}+10c - 7\). First, find two numbers that multiply to \(8\times(-7)=-56\) and add up to \(10\). The numbers are \(14\) and \(- 4\) since \(14\times(-4)=-56\) and \(14+( - 4)=10\).
Rewrite the middle term: \(8c^{2}+14c-4c - 7\)

Step2: Group and factor

Group the first two terms and the last two terms: \((8c^{2}+14c)+(-4c - 7)\)
Factor out the GCF from each group: \(2c(4c + 7)-1(4c + 7)\)
Now, factor out the common binomial factor \((4c + 7)\): \((2c - 1)(4c + 7)\)
Wait, maybe there was a typo in the options? Wait, let's check again. Wait, maybe I made a mistake. Wait, let's try another way. Let's check the options by plugging in the roots. For a factor \((ax + b)\), the root is \(c=-\frac{b}{a}\). Let's check option A: \((4c + 1)\), root \(c=-\frac{1}{4}\). Plug into \(8c^{2}+10c - 7\): \(8\times(-\frac{1}{4})^{2}+10\times(-\frac{1}{4})-7=8\times\frac{1}{16}-\frac{10}{4}-7=\frac{1}{2}-\frac{5}{2}-7=-2 - 7=-9 eq0\). Wait, maybe the options were miswritten? Wait, maybe the original problem has different options. Wait, perhaps the user made a typo, but assuming the correct factoring, let's re - do the AC method. Wait, \(8c^{2}+10c - 7\), \(a = 8\), \(b = 10\), \(c=-7\). The discriminant is \(b^{2}-4ac=100-4\times8\times(-7)=100 + 224 = 324=18^{2}\). The roots are \(c=\frac{-10\pm18}{16}\). So \(c=\frac{-10 + 18}{16}=\frac{8}{16}=\frac{1}{2}\) and \(c=\frac{-10-18}{16}=\frac{-28}{16}=-\frac{7}{4}\). So the factors are \((c-\frac{1}{2})\) and \((c+\frac{7}{4})\), multiplying by \(2\) and \(4\) to clear fractions: \(2(c-\frac{1}{2})=2c - 1\) and \(4(c+\frac{7}{4})=4c + 7\). So the factored form is \((2c - 1)(4c + 7)\). But since the options given seem to have typos, but if we assume that maybe option A was supposed to be \((4c + 7)\) or another form, but based on the given options (even with the extra text), if we check the calculation again, maybe there was a mistake in my first step. Wait, let's re - factor:

Wait, \(8c^{2}+10c - 7\). Let's try to factor as \((4c + a)(2c + b)\). Then \(4c\times2c = 8c^{2}\), \(4c\times b+2c\times a=10c\), and \(a\times b=-7\). We need \(4b + 2a = 10\) or \(2b + a = 5\) and \(ab=-7\). The factors of \(-7\) are \(7\) and \(-1\) or \(-7\) and \(1\). Let's try \(a = 7\), \(b=-1\): \(2\times(-1)+7 = 5\). Yes! So \((4c + 7)(2c - 1)\). So the factors are \((4c + 7)\) and \((2c - 1)\). If we assume that option A was a typo and should be \((4c + 7)\), but based on the given option A \((4c + 1)\) is not a factor. Wait, maybe the user made a mistake in the options. But if we proceed with the calculation, the correct factors are \((2c - 1)\) and \((4c + 7)\).

But since the options are given as A) \((4c + 1)\) (with extra text) and B) \((c - 7)\) (with extra text), maybe there was a mistake in the problem presentation. However, if we check the calculation again, the correct factoring gives \((2c - 1)(4c + 7)\). So if we assume that maybe the option A was supposed to be \((4c + 7)\), but as per the given options, there might be an error. But perhaps the original problem had different options.

Wait, maybe I made a mistake in the AC method. Let's try again. \(8c^{2}+10c - 7\). Multiply \(a = 8\) and \(c=-7\) to get \(-56\). Find two numbers that multiply to \(-56\) and add to \(10\). The numbers are \(14\) and \(-4\). So rewrite the middle term: \(8c^{2}+14c-4c - 7\). Factor by grouping: \(2c(4c + 7)-1(4c + 7)=(2c - 1)(4c + 7)\). So the factors are \((2c - 1)\) and \((4c + 7)\).

Since the options given don't match exactly, but if we consider that maybe there was a typo and option A was \((4c + 7)\), then that would be a factor. But based on the given option A \((4c + 1)\), it's not a factor. Wait, maybe the user made a mistake in the problem. But assuming that the intended answer is related to the factoring, and if we consider the options, maybe there was a typo and the correct factor is \((4c + 7)\) or \((2c - 1)\). But since the options are as given, perhaps there is an error. However, if we proceed with the calculation, the correct factors are \((2c - 1)\) and \((4c + 7)\).

Answer:

Since there seems to be a typo in the options, but if we assume that option A was supposed to be \((4c + 7)\) (correcting the typo), then the factor is \(\boldsymbol{(4c + 7)}\), but based on the given option A \((4c + 1)\) is not a factor. If we consider the correct factoring, the factors are \((2c - 1)\) and \((4c + 7)\).