Factor the Expression: Decomposing 4a+13b+58c Step by Step

Q: Why can't I factor out 2 from this expression?

While 2 divides 4 and 58, it doesn't divide 13. For factoring, your common factor must divide all coefficients. Since $ 13 ÷ 2 = 6.5 $ (not a whole number), 2 isn't a valid common factor.

Q: What makes 13 special in this problem?

13 is a prime number, meaning its only factors are 1 and 13 itself. Since neither 4 nor 58 is divisible by 13, there's no way to factor out 13 from all three terms.

Q: How do I check if numbers have a common factor?

Find the Greatest Common Divisor (GCD) of all coefficients. List the factors of each number, then find the largest number that appears in all lists. If the GCD is 1, no factoring is possible.

Q: Can expressions with different variables ever be factored?

Yes, but only if they share a common factor! For example, $ 6a + 12b + 18c $ can be factored as $ 6(a + 2b + 3c) $ because 6 divides all coefficients.

Q: What should I do when factoring isn't possible?

Simply state that "the expression cannot be factored" or "no common factor exists". This is a perfectly valid mathematical conclusion - not every expression can be factored!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's break this down into factors. Ready?

00:10 Don't worry, it looks like there are no common factors here.

00:19 And there you have it! We've solved the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Decompose the following expression into factors:

$4a+13b+58c$

2

Step-by-step solution

Factor the given expression:

$4a+13b+58c$ We will do this by extracting the greatest common factor, both from the numbers and the letters,

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

Let's start with the numbers:

We first notice that the numerical coefficients of the terms in the given expression, that is, the numbers 4, 13, 58, do not have a common factor, and this is because the number 13 is a prime number and the other two numbers are not multiples of it,

Therefore, there is no common factor for the numbers hence we select the number 1, as the common factor for the numbers (Reminder: a number raised to the power zero is always given to be equal to one)

For the letters:

There are three terms in the expression:
$a,\hspace{4pt}b,\hspace{4pt}c$ It is easy to see that there is no common factor for these three terms,

Hence, it is not possible to factor the given expression with the help of a common factor.

Therefore, the correct answer is option d.

3

Final Answer

It is not possible to factorize the given expression by extracting the common factor.

Key Points to Remember

Essential concepts to master this topic

Rule: Common factors must divide all coefficients in the expression
Technique: Check if 13 divides 4 and 58: 4÷13 and 58÷13 don't give whole numbers
Check: When no common factor exists, the expression cannot be factored ✓

Common Mistakes

Avoid these frequent errors

Forcing factorization when no common factor exists
Don't try to factor $4a+13b+58c$ by using partial coefficients like 2 or artificial groupings = incorrect mathematical manipulation! This creates false factors that don't actually divide all terms. Always verify that your proposed factor divides every single coefficient completely.

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 2 from this expression?

+

While 2 divides 4 and 58, it doesn't divide 13. For factoring, your common factor must divide all coefficients. Since $13 ÷ 2 = 6.5$ (not a whole number), 2 isn't a valid common factor.

What makes 13 special in this problem?

+

13 is a prime number, meaning its only factors are 1 and 13 itself. Since neither 4 nor 58 is divisible by 13, there's no way to factor out 13 from all three terms.

How do I check if numbers have a common factor?

+

Find the Greatest Common Divisor (GCD) of all coefficients. List the factors of each number, then find the largest number that appears in all lists. If the GCD is 1, no factoring is possible.

Can expressions with different variables ever be factored?

+

Yes, but only if they share a common factor! For example, $6a + 12b + 18c$ can be factored as $6(a + 2b + 3c)$ because 6 divides all coefficients.

What should I do when factoring isn't possible?

+

Simply state that "the expression cannot be factored" or "no common factor exists". This is a perfectly valid mathematical conclusion - not every expression can be factored!

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