Solve the Binomial Expression: Expanding (4a-b)(b+3a)

Q: Why do I need to multiply every term by every other term?

The distributive property requires each term in the first binomial to multiply with every term in the second binomial. Think of it like $ (a+b)(c+d) = ac + ad + bc + bd $ - no shortcuts allowed!

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

Use the FOIL method for binomials: First, Outer, Inner, Last. Or draw lines connecting each term in the first parentheses to each term in the second parentheses.

Q: What if I get confused with the negative signs?

Write out each step carefully! When you see $ -b \times b $, remember that negative times positive equals negative: $ -b^2 $.

Q: How do I combine like terms correctly?

Look for terms with the same variables and exponents. In this problem, $ 4ab $ and $ -3ab $ are like terms because they both have ab. Combine: $ 4ab - 3ab = ab $.

Q: Why isn't my final answer matching the choices?

Double-check your distribution and combining steps. Make sure you distributed the negative sign correctly and combined all like terms. The final answer should be $ 12a^2 - b^2 + ab $.

Binomial Expansion with Mixed Terms

Solve the exercise:

$(4a-b)(b+3a)=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's explore the solution, step by step.

00:10 First, open the parentheses carefully. Multiply each term inside by each term outside.

00:29 Now, calculate each of these multiplications.

00:57 Remember, a positive number times a negative number always gives a negative result.

01:09 Next, arrange the expression and group similar terms together.

01:15 And there you have it! That's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the exercise:

$(4a-b)(b+3a)=$

Step-by-step solution

To solve this problem, we will expand the expression $(4a-b)(b+3a)$ using the distributive property:

Firstly, use the distributive property to expand:

Step 1: Distribute $4a$ across both terms in $(b + 3a)$ :
$4a \cdot b = 4ab$ and $4a \cdot 3a = 12a^2$
Step 2: Distribute $-b$ across both terms in $(b + 3a)$ :
$-b \cdot b = -b^2$ and $-b \cdot 3a = -3ab$

Combine all these terms:

$4ab + 12a^2 - b^2 - 3ab$

Combine like terms:

The terms $4ab$ and $-3ab$ combine to give $ab$ .

Thus, the simplified form of the expression is:

$12a^2 - b^2 + ab - ab = 12a^2 - b^2 - ab$

Therefore, the solution to the problem is $12a^2 - b^2 - ab$ , which corresponds to choice $2$ .

Final Answer

$12a^2-b^2-ab$

Key Points to Remember

Essential concepts to master this topic

Distribution: Each term in first binomial multiplies every term in second
Technique: $4a \times b = 4ab$ and $-b \times 3a = -3ab$
Check: Combine like terms: $4ab - 3ab = ab$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs
Don't ignore the negative sign when distributing $-b$ = wrong signs throughout! This changes $-b \times b = -b^2$ to positive $b^2$ and gives completely wrong answers. Always carefully distribute negative signs to every term.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why do I need to multiply every term by every other term?

The distributive property requires each term in the first binomial to multiply with every term in the second binomial. Think of it like $(a+b)(c+d) = ac + ad + bc + bd$ - no shortcuts allowed!

How do I keep track of all the multiplications?

Use the FOIL method for binomials: First, Outer, Inner, Last. Or draw lines connecting each term in the first parentheses to each term in the second parentheses.

What if I get confused with the negative signs?

Write out each step carefully! When you see $-b \times b$ , remember that negative times positive equals negative: $-b^2$ .

How do I combine like terms correctly?

Look for terms with the same variables and exponents. In this problem, $4ab$ and $-3ab$ are like terms because they both have ab. Combine: $4ab - 3ab = ab$ .

Why isn't my final answer matching the choices?

Double-check your distribution and combining steps. Make sure you distributed the negative sign correctly and combined all like terms. The final answer should be $12a^2 - b^2 + ab$ .

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