which expression is a factor of $30x^{2}-4x - 16$?\na $5x + 4$\nb $3x - 2$\nc $5x - 4$\nd $3x - 4$

Explanation:

Step1: Factor the quadratic expression

For a quadratic expression (ax^{2}+bx + c) (here (a = 30), (b=-4), (c = - 16)), we can use the AC - method. First, find (ac=30\times(-16)=-480). We need to find two numbers (m) and (n) such that (m\times n=-480) and (m + n=b=-4). The numbers are (m = 20) and (n=-24) since (20\times(-24)=-480) and (20+(-24)=-4). Rewrite the middle term: (30x^{2}+20x-24x - 16) Group the terms: ((30x^{2}+20x)-(24x + 16)) Factor out the GCF from each group: (10x(3x + 2)-8(3x + 2)) Then ((3x + 2)(10x-8)=2(3x + 2)(5x - 4))

Step2: Check the options

We can also use the factor theorem. For a binomial (ax + b), if (x=-\frac{b}{a}) is a root of (P(x)=30x^{2}-4x-16), then (ax + b) is a factor of (P(x)). For option C: Let (P(x)=30x^{2}-4x - 16) and consider the binomial (5x-4), then (x=\frac{4}{5}) (P(\frac{4}{5})=30\times(\frac{4}{5})^{2}-4\times\frac{4}{5}-16) (=30\times\frac{16}{25}-\frac{16}{5}-16) (=\frac{480}{25}-\frac{80}{25}-\frac{400}{25}=\frac{480 - 80-400}{25}=0)

Answer:

C. (5x - 4)