Question

error analysis your friend incorrectly factors the expression below as 7x(5 - 2xy). factor the expression below correctly. what error did your friend make?
35x - 14xy
factor the expression.
35x - 14xy = \square (factor completely)

Explanation:

Step1: Find GCF of coefficients and variables

The GCF of 35 and 14 is 7. For variables, both terms have \(x\), so GCF of variables is \(x\). Thus, GCF is \(7x\).

Step2: Divide each term by GCF

Divide \(35x\) by \(7x\): \(\frac{35x}{7x} = 5\). Divide \(-14xy\) by \(7x\): \(\frac{-14xy}{7x} = -2y\).

Step3: Write factored form

Using distributive property, \(35x - 14xy = 7x(5 - 2y)\). Wait, but the friend had \(7x(5 - 2xy)\), so the error was in dividing \(-14xy\) by \(7x\) (friend got \(2xy\) instead of \(2y\)). But for correct factoring, it's \(7x(5 - 2y)\)? Wait, no, wait original expression: \(35x - 14xy\). Wait, \(35x = 7x\times5\), \(-14xy = 7x\times(-2y)\). So factored form is \(7x(5 - 2y)\). Wait, but the friend's answer was \(7x(5 - 2xy)\), so the mistake was that when factoring out \(7x\) from \(-14xy\), they should get \(-2y\) not \(-2xy\) (they incorrectly kept an extra \(x\)). But the question is to factor \(35x - 14xy\) completely. So let's do it again:

GCF of \(35x\) and \(14xy\): prime factors of 35: 5×7, 14: 2×7. Variables: \(x\) is common. So GCF is \(7x\). Then \(35x\div7x = 5\), \( -14xy\div7x = -2y\). So factored form is \(7x(5 - 2y)\). Wait, but let's check: \(7x\times5 = 35x\), \(7x\times(-2y) = -14xy\). Correct. So the friend's error was in the second term's factorization (they had \(-2xy\) instead of \(-2y\)). But the question is to factor \(35x - 14xy\) completely, so the correct factoring is \(7x(5 - 2y)\)? Wait, no, wait the original problem's expression: is it \(35x - 14xy\)? Let me check the image again. The image says "35x - 14xy". So yes. So factoring:

\(35x - 14xy = 7x(5 - 2y)\). Wait, but maybe I made a mistake. Wait, \(35x = 5\times7x\), \(14xy = 2\times7x\times y\). So factoring out \(7x\), we get \(7x(5 - 2y)\). So the friend's error was that when dividing \(-14xy\) by \(7x\), they got \(-2xy\) instead of \(-2y\) (they incorrectly retained an \(x\) in the second term). But the correct factored form is \(7x(5 - 2y)\). Wait, but let's verify:

\(7x(5 - 2y) = 35x - 14xy\), which matches the original expression. So that's correct.

Answer:

\(7x(5 - 2y)\)