Question

polynomials operations - review for test#2 (adding, subtracting, multiplying)\
which expression is equivalent to $10cd^2 + 6c^2d - 12d$?\
$\circ$ $2c(5cd + 3c - 6)$\
$\circ$ $2d(5cd + 3c - 6)$\
$\circ$ $2cd(5cd + 3c - 6)$\
$\circ$ $2(5cd - 3c + 6)$

Explanation:

Step1: Analyze the original expression

The original expression is \(10cd^2 + 6c^2d - 12d\). We need to factor out the greatest common factor (GCF) from each term. First, find the GCF of the coefficients \(10\), \(6\), and \(-12\). The GCF of \(10\), \(6\), and \(12\) is \(2\). Then, look at the variables: the common variable terms are \(d\) (since \(d\) appears in all terms, with the lowest power of \(1\) in \(10cd^2\) (wait, no, \(10cd^2\) has \(d^2\), \(6c^2d\) has \(d^1\), \(-12d\) has \(d^1\)). Wait, maybe I misread the original expression. Wait, the problem's original expression: let's re - examine. Wait, the user's image: the original expression is \(10cd^2+6c^2d - 12d\)? Wait, no, maybe the original expression is \(10cd^2 + 6c^2d-12d\)? Wait, no, looking at the options, let's check the options. The options are of the form \(2d(5cd + 3c - 6)\), \(2c(5cd + 3c - 6)\), \(2d(5cd + 3c - 6)\), \(2cd(5cd + 3c - 6)\), \(2(5cd - 3c + 6)\). Wait, let's factor the original expression. Let's assume the original expression is \(10cd^2+6c^2d - 12d\). Factor out \(2d\) from each term: \(2d\times5cd+2d\times3c - 2d\times6=2d(5cd + 3c - 6)\). Wait, let's check the coefficients: \(10\div2 = 5\), \(6\div2=3\), \(- 12\div2=-6\). And the variables: \(10cd^2=2d\times5cd\), \(6c^2d = 2d\times3c\), \(-12d=2d\times(-6)\). So the factored form is \(2d(5cd + 3c - 6)\), which is one of the options (the third option: \(2d(5cd + 3c - 6)\)).

Step2: Verify by expanding the factored form

Let's expand \(2d(5cd + 3c - 6)\). Using the distributive property \(a(b + c + d)=ab+ac + ad\), where \(a = 2d\), \(b = 5cd\), \(c = 3c\), \(d=-6\). So \(2d\times5cd=10c d^{2}\), \(2d\times3c = 6c^{2}d\), \(2d\times(-6)=-12d\). Which matches the original expression (assuming the original expression is \(10cd^2+6c^2d - 12d\)).

Answer:

The equivalent expression is \(2d(5cd + 3c - 6)\) (the third option in the list of options provided, i.e., the option with text \(2d(5cd + 3c - 6)\)).