Question

which two expressions are equivalent?
a $x + x + x$ and $x^3$

b $14x + 10 - 2x$ and $16x + 10$

c $12x + 16x$ and $4(3x + 4x)$

d $12x^2 + 5x + 10$ and $17x^2 + 10$

a

b

c

Explanation:

Step1: Analyze Option A

Simplify \(x + x + x\): combining like terms, we get \(3x\). \(x^3=x\times x\times x\), which is not equal to \(3x\) (e.g., if \(x = 2\), \(3x=6\), \(x^3 = 8\)). So A is incorrect.

Step2: Analyze Option B

Simplify \(14x+10 - 2x\): combine like terms \(14x-2x = 12x\), so the expression becomes \(12x + 10\), which is not equal to \(16x+10\) (e.g., \(x = 1\), \(12x + 10=22\), \(16x + 10 = 26\)). So B is incorrect.

Step3: Analyze Option C

Simplify \(12x+16x\): combine like terms, \(12x + 16x=28x\). Simplify \(4(3x + 4x)\): first simplify inside the parentheses, \(3x+4x = 7x\), then multiply by 4: \(4\times7x = 28x\). Both expressions equal \(28x\), so they are equivalent.

Step4: Analyze Option D (Optional, since we found C is correct)

\(12x^2+5x + 10\) has a \(5x\) term and \(12x^2\), while \(17x^2+10\) has \(17x^2\) and no \(x\) term. They are not equivalent (e.g., \(x = 1\), \(12x^2+5x + 10=27\), \(17x^2+10 = 27\)? Wait, no, \(12(1)^2+5(1)+10=12 + 5+10 = 27\), \(17(1)^2+10=17 + 10 = 27\)? Wait, no, that's a coincidence. Wait, \(x = 2\): \(12(4)+10 + 10=48 + 10+10=68\)? Wait, no, original D: \(12x^2+5x + 10\) and \(17x^2+10\). For \(x = 2\), \(12(4)+10 + 10=48+10 + 10=68\)? No, \(12x^2+5x + 10=12(4)+10 + 10=48 + 5+10=63\), \(17x^2+10=17(4)+10=68 + 10=78\). So they are not equivalent. So D is incorrect.

Answer:

C. \(12x + 16x\) and \(4(3x + 4x)\)