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

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

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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