Find the Missing Term in (3x-8)(-4+?) = -3x²-4x+32

Q: Why does the missing term create an x² term?

When you multiply $ 3x $ by the missing term $ ? $, you get $ 3x \cdot ? $. If $ ? = -x $, then $ 3x \cdot (-x) = -3x^2 $, which matches our target expression!

Q: How do I know which terms to compare?

After expanding, group terms by their degree: all $ x^2 $ terms together, all $ x $ terms together, and all constants together. Then match coefficients with the given polynomial.

Q: What if I get confused during the expansion?

Use the FOIL method or distributive property step by step: $ (3x-8)(-4+?) = 3x(-4) + 3x(?) + (-8)(-4) + (-8)(?) $. Take it one multiplication at a time!

Q: Can the missing term be a number instead of a variable?

Look at the target expression! Since we need $ -3x^2 $, and this comes from $ 3x \cdot ? $, the missing term must contain x. A plain number couldn't create the $ x^2 $ term.

Q: How can I check my answer quickly?

Substitute your answer back: $ (3x-8)(-4+(-x)) = (3x-8)(-4-x) $. Expand this and verify it equals $ -3x^2-4x+32 $!

Binomial Expansion with Missing Terms

Complete the missing element

$(3x-8)(-4+?)=-3x^2-4x+32$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's find the missing term.

00:15 Use Y as our unknown. Great job so far!

00:20 Now, open those parentheses. Multiply each number by each other. Take your time!

00:41 Next, calculate the products. You're doing well!

00:56 Now, reduce where you can and gather like terms. Almost there!

01:21 Factor out any common terms on both sides. Keep going!

01:30 Isolate Y, our unknown.

01:36 And that's how we solve the problem! Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the missing element

$(3x-8)(-4+?)=-3x^2-4x+32$

Step-by-step solution

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

Step 1: Expand the binomial expression using the distributive property.
Step 2: Compare the resulting polynomial's coefficients with the given expression's coefficients.
Step 3: Solve for the missing element.

Now, let's work through each step:
Step 1: Express $(3x-8)(-4+?)$ using the distributive property:
$(3x)(-4) + (3x)(?) - (8)(-4) - (8)(?)$ .
This simplifies to: $-12x + 3x(?) + 32 - 8(?)$ .

Step 2: Compare with $-3x^2 - 4x + 32$ , equaling terms by degree:
- The constant term (32) already matches.
- The $x$ term is $-12x + 3x(?) = -4x$ .
- The $x^2$ term arises from $3x(?)$ .

Step 3: Solve for the missing element by aligning coefficients:
$3x(?) = -3x^2$ , therefore $? = -x$ .
Thus, the missing element is $-x$ .

Therefore, the solution to the problem is $-x$ , which corresponds to choice 2.

Final Answer

$-x$

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Expand (3x-8)(-4+?) by multiplying each term
Coefficient Matching: Compare x², x, and constant terms: 3x(?) = -3x²
Verification: Substitute ? = -x back into original equation to confirm ✓

Common Mistakes

Avoid these frequent errors

Ignoring the x² term when comparing coefficients
Don't focus only on the x and constant terms = missing the key insight! The x² term comes from multiplying 3x by the missing term, so 3x(?) = -3x² means ? = -x. Always analyze all polynomial terms systematically.

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 does the missing term create an x² term?

When you multiply $3x$ by the missing term $?$ , you get $3x \cdot ?$ . If $? = -x$ , then $3x \cdot (-x) = -3x^2$ , which matches our target expression!

How do I know which terms to compare?

After expanding, group terms by their degree: all $x^2$ terms together, all $x$ terms together, and all constants together. Then match coefficients with the given polynomial.

What if I get confused during the expansion?

Use the FOIL method or distributive property step by step: $(3x-8)(-4+?) = 3x(-4) + 3x(?) + (-8)(-4) + (-8)(?)$ . Take it one multiplication at a time!

Can the missing term be a number instead of a variable?

Look at the target expression! Since we need $-3x^2$ , and this comes from $3x \cdot ?$ , the missing term must contain x. A plain number couldn't create the $x^2$ term.

How can I check my answer quickly?

Substitute your answer back: $(3x-8)(-4+(-x)) = (3x-8)(-4-x)$ . Expand this and verify it equals $-3x^2-4x+32$ !

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