Factorize the Expression: a²b²c² - a²b⁵c⁴ + a⁻⁴b³c⁷

Q: How do I divide terms with negative exponents?

Use the rule: $ a^m ÷ a^n = a^{m-n} $. So $ a^{-4} ÷ a^2 = a^{-4-2} = a^{-6} $. Remember: subtracting a positive is the same as adding a negative!

Q: Can I factor this expression differently?

Yes, but factoring out $ a^2b^2c^2 $ gives the simplest form with manageable terms inside the parentheses. Other factorizations would be more complex.

Q: Why does the first term become 1 after factoring?

Because $ \frac{a^2b^2c^2}{a^2b^2c^2} = 1 $! Any expression divided by itself equals 1. This is why the first term in the parentheses is simply 1.

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

Multiply out your factored form: $ a^2b^2c^2(1 - b^3c^2 + a^{-6}bc^5) $. Distribute to get back the original three terms. If they match exactly, you're correct! ✓

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Decompose the following expression into factors:

$a^2b^2c^2-a^2b^5c^4+a^{-4}b^3c^7$

2

Step-by-step solution

To solve this problem, let's perform the factorization step by step:

We start with the expression:

a^2b^2c^2 - a^2b^5c^4 + a^{-4}b^3c^7

Step 1: Identify the common factors among terms:

The smallest exponent of $a$ among terms with positive exponents is $2$ , excluding any negative exponent (as considering $a^{-4}$ would introduce complexity).
The smallest exponent of $b$ is $2$ .
The smallest exponent of $c$ is $2$ .

Therefore, a common factor considering positive indices is:

a^2b^2c^2

Step 2: Factor out $a^2b^2c^2$ from the entire expression:

a^2b^2c^2 \left( \frac{a^2b^2c^2}{a^2b^2c^2} - \frac{a^2b^5c^4}{a^2b^2c^2} + \frac{a^{-4}b^3c^7}{a^2b^2c^2} \right)

Simplifying the terms inside the parenthesis, we have:

$\frac{a^2b^2c^2}{a^2b^2c^2} = 1$ (since dividing any expression by itself gives 1).
$\frac{a^2b^5c^4}{a^2b^2c^2} = b^3c^2$ .
$\frac{a^{-4}b^3c^7}{a^2b^2c^2} = a^{-6}bc^5$ .

Putting it all together inside the parenthesis, we get:

1 - b^3c^2 + a^{-6}bc^5

Thus, the completely factored expression is:

a^2b^2c^2 (1 - b^3c^2 + a^{-6}bc^5)

Verifying against given choices, this matches exactly with the correct choice:

$a^2b^2c^2\left(a-b^3c^3+a^{-6}bc^5\right)$

Therefore, the solution to the problem is $\boxed{a^2b^2c^2(a-b^3c^3+a^{-6}bc^5)}$ .

3

Final Answer

$a^2b^2c^2\left(a-b^3c^3+a^{-6}bc^5\right)$

Key Points to Remember

Essential concepts to master this topic

Common Factor: Find the smallest positive exponent for each variable
Division Rule: $\frac{a^{-4}b^3c^7}{a^2b^2c^2} = a^{-6}bc^5$
Verification: Expand the factored form to get original expression ✓

Common Mistakes

Avoid these frequent errors

Including negative exponents in common factor
Don't factor out $a^{-4}$ as the common factor = creates more complex terms! This makes the expression harder to work with. Always use the smallest positive exponent when choosing common factors.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why don't we use a⁻⁴ as the common factor since it's the smallest exponent?

+

Using negative exponents as common factors makes the problem more complicated, not simpler! We factor out $a^2b^2c^2$ to keep the remaining terms manageable.

How do I divide terms with negative exponents?

+

Use the rule: $a^m ÷ a^n = a^{m-n}$ . So $a^{-4} ÷ a^2 = a^{-4-2} = a^{-6}$ . Remember: subtracting a positive is the same as adding a negative!

Can I factor this expression differently?

+

Yes, but factoring out $a^2b^2c^2$ gives the simplest form with manageable terms inside the parentheses. Other factorizations would be more complex.

Why does the first term become 1 after factoring?

+

Because $\frac{a^2b^2c^2}{a^2b^2c^2} = 1$ ! Any expression divided by itself equals 1. This is why the first term in the parentheses is simply 1.

How do I check if my factorization is correct?

+

Multiply out your factored form: $a^2b^2c^2(1 - b^3c^2 + a^{-6}bc^5)$ . Distribute to get back the original three terms. If they match exactly, you're correct! ✓

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Factorize the Expression: a²b²c² - a²b⁵c⁴ + a⁻⁴b³c⁷

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