Expand (a+4b)(2c+a): Binomial Multiplication with Three Variables

Q: Why do I get four terms instead of two?

When multiplying two binomials, you get four terms because each term from the first binomial multiplies with each term from the second. Think $ (a+4b)(2c+a) $ as distributing completely!

Q: How do I keep track of all the multiplications?

Use the FOIL method or draw connecting lines: First terms, Outside terms, Inside terms, Last terms. For $ (a+4b)(2c+a) $, that's $ a \cdot 2c $, $ a \cdot a $, $ 4b \cdot 2c $, $ 4b \cdot a $.

Q: Can I combine any of these terms?

Look for like terms - terms with the same variables and exponents. In $ 2ac + a^2 + 8bc + 4ab $, each term is different, so no combining is possible here!

Q: Does the order of terms in my answer matter?

No! $ 2ac + a^2 + 8bc + 4ab $ is the same as $ a^2 + 2ac + 4ab + 8bc $. However, it's conventional to write terms in alphabetical order or by degree.

Q: What if I have negative signs in the binomials?

Follow the same distribution process, but be extra careful with signs! Remember that negative times positive equals negative, and negative times negative equals positive.

Binomial Multiplication with Three Variables

$(a+4b)(2c+a)=$

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Step-by-step video solution

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00:00 Solve

00:04 Open parentheses properly, multiply each factor by each factor

00:26 Calculate the products

00:45 Arrange the expression

00:51 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

$(a+4b)(2c+a)=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Distribute each term in the first binomial $(a + 4b)$ to each term in the second binomial $(2c + a)$ .
Step 2: Simplify the expanded terms by combining like terms (if any).

Now, let's work through each step:
Step 1: Distribute $a$ from the first binomial to each term in the second binomial: - Distribute $a$ to $2c$ : $a \cdot 2c = 2ac$ - Distribute $a$ to $a$ : $a \cdot a = a^2$
Step 2: Distribute $4b$ from the first binomial to each term in the second binomial: - Distribute $4b$ to $2c$ : $4b \cdot 2c = 8bc$ - Distribute $4b$ to $a$ : $4b \cdot a = 4ab$
Now, combining all these results gives us:
$2ac + a^2 + 8bc + 4ab$

Therefore, the expanded form of the expression $(a+4b)(2c+a)$ is $2ac + a^2 + 8bc + 4ab$ .

Final Answer

$2ac+a^2+8bc+4ab$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: Distribute each term from first binomial to each term in second
Technique: $a \cdot 2c = 2ac$ and $4b \cdot a = 4ab$
Check: Count four terms in final answer: $2ac + a^2 + 8bc + 4ab$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute to all terms
Don't multiply just the first terms together = missing three terms! This gives incomplete answers like $2ac$ instead of all four terms. Always ensure each term in the first binomial multiplies with each term in the second binomial.

Practice Quiz

Test your knowledge with interactive questions

$(3+20)\times(12+4)=$

FAQ

Everything you need to know about this question

Why do I get four terms instead of two?

When multiplying two binomials, you get four terms because each term from the first binomial multiplies with each term from the second. Think $(a+4b)(2c+a)$ as distributing completely!

How do I keep track of all the multiplications?

Use the FOIL method or draw connecting lines: First terms, Outside terms, Inside terms, Last terms. For $(a+4b)(2c+a)$ , that's $a \cdot 2c$ , $a \cdot a$ , $4b \cdot 2c$ , $4b \cdot a$ .

Can I combine any of these terms?

Look for like terms - terms with the same variables and exponents. In $2ac + a^2 + 8bc + 4ab$ , each term is different, so no combining is possible here!

Does the order of terms in my answer matter?

No! $2ac + a^2 + 8bc + 4ab$ is the same as $a^2 + 2ac + 4ab + 8bc$ . However, it's conventional to write terms in alphabetical order or by degree.

What if I have negative signs in the binomials?

Follow the same distribution process, but be extra careful with signs! Remember that negative times positive equals negative, and negative times negative equals positive.

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