Factor the Expression: Decomposing xyz+yzt+ztw+wtr Using Common Factors

Q: Why can't I factor out 'z' since it appears in three terms?

A common factor must appear in every single term. Since 'z' is missing from the term $ wtr $, it cannot be factored out. All four terms must contain the same variable for factoring to work.

Q: How do I know for certain that an expression cannot be factored?

Check systematically: list all variables in each term. If no variable appears in every term, then no common factor exists. For $ xyz+yzt+ztw+wtr $, no single variable is present in all four terms.

Q: Is it wrong to say an expression cannot be factored?

Not at all! Recognizing when factoring is impossible is just as important as knowing how to factor. This shows you understand the requirements for factoring and won't waste time on impossible problems.

Factoring Polynomials with No Common Variables

Decompose the following expression into factors by removing the common factor:

$xyz+yzt+ztw+wtr$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Break down into factors

00:03 Mark the common factors

00:13 We see there are no common factors

00:19 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Decompose the following expression into factors by removing the common factor:

$xyz+yzt+ztw+wtr$

Step-by-step solution

Factor the given expression:

$xyz+yzt+ztw+wtr$
We will do this by extracting the highest common factor, both from the numbers and the letters.

We refer to the numbers and letters separately, remembering that a common factor is a factor (multiplier) common to all terms of the expression.

As the given expression does not have numeric coefficients (other than 1), we will look for the highest common factor of the letters:

There are four terms in the expression:
$xyz,\hspace{4pt}yzt,\hspace{4pt}ztw,\hspace{4pt}wtr$ We will notice that in each of the four members there are three different letters, but there is not one or more letters that are included (in the multiplication) in all the terms; that is, there is no common factor for the four terms and therefore it is not possible to factor this expression by extracting a common factor.

Therefore, the correct answer is option d.

Final Answer

It is not possible to decompose the given expression into factors by extracting the common factor.

Key Points to Remember

Essential concepts to master this topic

Common Factor Rule: All terms must share the same variable(s)
Analysis Method: Check each variable - xyz has x, but wtr doesn't
Verification: If no variable appears in every term, factoring is impossible ✓

Common Mistakes

Avoid these frequent errors

Forcing factorization when no common factor exists
Don't try to factor by grouping random terms like xy(z+zt+wt+rw) = wrong factors! This creates terms that weren't in the original expression. Always verify that every variable appears in ALL terms before attempting to factor.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 2x^2 $$

FAQ

Everything you need to know about this question

Why can't I factor out 'z' since it appears in three terms?

A common factor must appear in every single term. Since 'z' is missing from the term $wtr$ , it cannot be factored out. All four terms must contain the same variable for factoring to work.

What about grouping terms differently to find factors?

Grouping only works if you can factor each group and find a common binomial factor. In this expression, no grouping produces the same factor in different groups, so factoring by grouping won't work either.

How do I know for certain that an expression cannot be factored?

Check systematically: list all variables in each term. If no variable appears in every term, then no common factor exists. For $xyz+yzt+ztw+wtr$ , no single variable is present in all four terms.

Are there other ways to factor this type of expression?

Not by extracting common factors. Some expressions might factor using special patterns like difference of squares, but this particular expression has no recognizable pattern that allows factoring.

Is it wrong to say an expression cannot be factored?

Not at all! Recognizing when factoring is impossible is just as important as knowing how to factor. This shows you understand the requirements for factoring and won't waste time on impossible problems.

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