Factorize the Expression: Breaking Down 26a + 65bc Step-by-Step

Q: How do I find the greatest common factor of two numbers?

Find the prime factorization of each number first. For 26: $ 26 = 2 \times 13 $. For 65: $ 65 = 5 \times 13 $. The GCF is the product of common prime factors, which is 13.

Q: What if the terms have different variables like 'a' and 'bc'?

No problem! When terms have different variables, focus only on the numerical coefficients. You can still factor out the GCF of the numbers (26 and 65), which is 13.

Q: How do I check if my factorization is correct?

Use the distributive property to expand your answer. For $ 13(2a + 5bc) $: multiply 13 by each term inside to get $ 13 \times 2a + 13 \times 5bc = 26a + 65bc $ ✓

Q: Why can't I factor out 26 or 65 instead of 13?

You want the greatest common factor, not just any common factor. While 26 divides the first term, it doesn't divide 65. Similarly, 65 doesn't divide 26. Only 13 divides both coefficients completely.

Factorization with Common Factors

Factorise:

$26a+65bc$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the common factor

00:07 Factor 26 into 13 and 2

00:11 Factor 65 into 13 and 5

00:16 Mark the common factors

00:25 Take out the common factors from the parentheses

00:35 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

Factorise:

$26a+65bc$

Step-by-step solution

To factor the expression $26a + 65bc$ , we start by identifying the greatest common factor of the coefficients:

The coefficients are 26 and 65.
The prime factorization of 26 is $2 \times 13$ .
The prime factorization of 65 is $5 \times 13$ .
The greatest common factor of 26 and 65 is 13, because both coefficients are divisible by 13.

Now, factor out $13$ from both terms of the expression:

$26a = 13 \times 2a$

$65bc = 13 \times 5bc$

So, we can express the entire expression as:

$26a + 65bc = 13(2a + 5bc)$

Therefore, the factorized form of the expression is $\mathbf{13(2a + 5bc)}$ .

Comparing with the given choices, this corresponds to choice 1.

Final Answer

$13(2a+5bc)$

Key Points to Remember

Essential concepts to master this topic

Greatest Common Factor: Find GCF by comparing prime factorizations of coefficients
Prime Factorization: 26 = 2×13, 65 = 5×13, so GCF = 13
Verification: Expand 13(2a + 5bc) = 26a + 65bc matches original ✓

Common Mistakes

Avoid these frequent errors

Trying to factor variables when there are no common variables
Don't look for variable factors like 'a' when terms don't share variables = impossible factorization! The expression 26a + 65bc has no common variables between terms. Always focus on coefficient factors first - find the GCF of numbers 26 and 65.

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

How do I find the greatest common factor of two numbers?

Find the prime factorization of each number first. For 26: $26 = 2 \times 13$ . For 65: $65 = 5 \times 13$ . The GCF is the product of common prime factors, which is 13.

What if the terms have different variables like 'a' and 'bc'?

No problem! When terms have different variables, focus only on the numerical coefficients. You can still factor out the GCF of the numbers (26 and 65), which is 13.

How do I check if my factorization is correct?

Use the distributive property to expand your answer. For $13(2a + 5bc)$ : multiply 13 by each term inside to get $13 \times 2a + 13 \times 5bc = 26a + 65bc$ ✓

Why can't I factor out 26 or 65 instead of 13?

You want the greatest common factor, not just any common factor. While 26 divides the first term, it doesn't divide 65. Similarly, 65 doesn't divide 26. Only 13 divides both coefficients completely.

What if there's no common factor between the coefficients?

If coefficients share no common factors (like 15 and 28), then the expression cannot be factored using common factors. The expression would already be in its simplest form.

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