Expand (mn-l)²: Converting to Addition and Multiplication Forms

Q: Why does (mn-l)² have three terms when expanded?

When you square a binomial, you get three terms: the square of the first term, twice the product of both terms (with proper sign), and the square of the second term. Think of it as (a-b)(a-b) using FOIL!

Q: How do I remember the binomial square formula?

Remember "First squared, twice the product, last squared". For (a-b)²: a² (first squared), -2ab (twice the product with minus), +b² (last squared).

Q: What's the difference between addition and multiplication forms?

The addition form shows the expanded trinomial (m²n² - 2mnl + l²). The multiplication form shows the original expression as a product: (mn-l)(mn-l).

Q: Can I just multiply (mn-l) by itself using FOIL?

Absolutely! Using FOIL on (mn-l)(mn-l) gives the same result as the binomial square formula. Both methods work perfectly - use whichever you find easier!

Q: Why is the middle term negative?

The middle term is negative because we have (mn - l)². When you multiply -l by mn twice (from FOIL), you get -mnl + (-mnl) = -2mnl.

Q: What if I had (mn + l)² instead?

With a plus sign, the formula becomes (a+b)² = a² + 2ab + b², so you'd get m²n² + 2mnl + l². The middle term would be positive!

Binomial Squares with Trinomial Expansion

Rewrite the expression below as an addition and as a multiplication:

$(mn-l)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the expression as a sum and as a product

00:06 First let's solve as a product

00:10 A factor times itself is actually squared

00:14 Let's use this formula and square the parentheses

00:25 Now let's solve as a sum

00:34 Let's use the shortened multiplication formulas to expand the parentheses

00:52 When squaring a product, each factor is squared

00:59 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

Rewrite the expression below as an addition and as a multiplication:

$(mn-l)^2$

Step-by-step solution

To solve this problem, we will first rewrite the given expression $(mn-l)^2$ using the binomial square formula.

The formula for the square of a binomial is:

$(a - b)^2 = a^2 - 2ab + b^2$

In the expression $(mn-l)^2$ , identify $a = mn$ and $b = l$ .

Apply the binomial square formula:

$(mn - l)^2 = (mn)^2 - 2(mn)(l) + l^2$

Calculate each term:

$(mn)^2 = m^2n^2$
$2(mn)(l) = 2mnl$
$l^2 = l^2$

Combine these to write the expression as a sum of terms:

$m^2n^2 - 2mnl + l^2$

Next, express $(mn-l)^2$ as a multiplication:

By definition, the square of a term can be rewritten as the term multiplied by itself:

$(mn-l)^2 = (mn-l)(mn-l)$

Therefore, the expression rewritten as an addition and a multiplication is:

$m^2n^2-2mnl+l^2$

$(mn-l)(mn-l)$

Final Answer

$m^2n^2-2mnl+l^2$

$(mn-l)(mn-l)$

Key Points to Remember

Essential concepts to master this topic

Formula: Use (a-b)² = a² - 2ab + b² for expansion
Technique: Identify a = mn and b = l, then calculate each term
Check: Multiply (mn-l)(mn-l) using FOIL to verify m²n² - 2mnl + l² ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term when expanding
Don't expand (mn-l)² as just m²n² + l² = incomplete trinomial! This misses the crucial -2mnl term from the cross products. Always remember (a-b)² = a² - 2ab + b² includes ALL three terms.

Practice Quiz

Test your knowledge with interactive questions

$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why does (mn-l)² have three terms when expanded?

When you square a binomial, you get three terms: the square of the first term, twice the product of both terms (with proper sign), and the square of the second term. Think of it as (a-b)(a-b) using FOIL!

How do I remember the binomial square formula?

Remember "First squared, twice the product, last squared". For (a-b)²: a² (first squared), -2ab (twice the product with minus), +b² (last squared).

What's the difference between addition and multiplication forms?

The addition form shows the expanded trinomial (m²n² - 2mnl + l²). The multiplication form shows the original expression as a product: (mn-l)(mn-l).

Can I just multiply (mn-l) by itself using FOIL?

Absolutely! Using FOIL on (mn-l)(mn-l) gives the same result as the binomial square formula. Both methods work perfectly - use whichever you find easier!

Why is the middle term negative?

The middle term is negative because we have (mn - l)². When you multiply -l by mn twice (from FOIL), you get -mnl + (-mnl) = -2mnl.

What if I had (mn + l)² instead?

With a plus sign, the formula becomes (a+b)² = a² + 2ab + b², so you'd get m²n² + 2mnl + l². The middle term would be positive!

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