Verify the Polynomial Equality: (b-3)(b+7) = b²+4b-21

Q: What does FOIL stand for and why use it?

FOIL stands for First, Outer, Inner, Last - the order you multiply terms in two binomials. It's a systematic way to ensure you don't miss any products when expanding $ (a+b)(c+d) $.

Q: How do I keep track of positive and negative signs?

Write out each step clearly! For $ (b-3)(b+7) $, the -3 stays negative in all its products: -3 × b = -3b and -3 × 7 = -21.

Q: Can I expand binomials in a different order?

Yes! You can multiply in any order, but FOIL helps you stay organized. Whether you do First-Outer-Inner-Last or group differently, you should get the same final answer.

Q: How do I verify my expansion is correct?

Substitute a simple value like b = 1 into both the original $ (b-3)(b+7) $ and your expanded form. If both give the same result, your expansion is likely correct!

Polynomial Expansion with FOIL Method

Is equality correct?

$(b-3)(b+7)=b^2+4b-21$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Are these expressions equal? Let's find out!

00:13 First, let's open the parentheses correctly. We'll multiply each factor by every other factor.

00:29 Now, let's work on calculating the products.

00:45 Great! Next, we'll group the factors together.

00:48 Let's compare the terms in the expressions to see if they match.

00:53 And that's how we find the solution to this question. Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Is equality correct?

$(b-3)(b+7)=b^2+4b-21$

Step-by-step solution

To solve this problem, let's expand and simplify the expression on the left-hand side:

$(b-3)(b+7)$

Applying the distributive property (FOIL), we have:

First: $b \times b = b^2$
Outer: $b \times 7 = 7b$
Inner: $-3 \times b = -3b$
Last: $-3 \times 7 = -21$

Combine these terms:

$b^2 + 7b - 3b - 21$

Simplify by combining like terms:

$b^2 + (7b - 3b) - 21 = b^2 + 4b - 21$

The expression on the left simplifies to $b^2 + 4b - 21$ . This is identical to the expression on the right-hand side of the equality.

Since both sides of the equation are equal, the given equality is correct.

Thus, the correct answer is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

FOIL Method: First, Outer, Inner, Last terms systematically expanded
Technique: $(b-3)(b+7) = b^2 + 7b - 3b - 21$
Verification: Combine like terms to match given expression perfectly ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the negative sign
Don't multiply -3 × 7 as positive 21 = wrong constant term! The negative sign belongs to the 3, so it affects all products with that term. Always keep track of signs: -3 × 7 = -21.

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

What does FOIL stand for and why use it?

FOIL stands for First, Outer, Inner, Last - the order you multiply terms in two binomials. It's a systematic way to ensure you don't miss any products when expanding $(a+b)(c+d)$ .

How do I keep track of positive and negative signs?

Write out each step clearly! For $(b-3)(b+7)$ , the -3 stays negative in all its products: -3 × b = -3b and -3 × 7 = -21.

What if I get different coefficients when combining like terms?

Double-check your FOIL work! Make sure you correctly identified the Outer and Inner terms. For $(b-3)(b+7)$ , Outer is b × 7 = 7b and Inner is -3 × b = -3b.

Can I expand binomials in a different order?

Yes! You can multiply in any order, but FOIL helps you stay organized. Whether you do First-Outer-Inner-Last or group differently, you should get the same final answer.

How do I verify my expansion is correct?

Substitute a simple value like b = 1 into both the original $(b-3)(b+7)$ and your expanded form. If both give the same result, your expansion is likely correct!

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