Verify if (3y+x)(4+2x) = 2x²+6xy+4x+12y: Polynomial Equality Check

Q: Do I have to multiply every term with every other term?

Yes, absolutely! When multiplying $ (3y+x)(4+2x) $, you need four multiplications: 3y×4, 3y×2x, x×4, and x×2x. Missing any step gives the wrong answer.

Q: Why does the order of terms matter when checking equality?

The order doesn't matter in polynomial equality! $ 12y + 6xy + 4x + 2x^2 $ equals $ 2x^2 + 6xy + 4x + 12y $ because addition is commutative.

Q: What's the easiest way to expand two polynomials?

Use the distributive property systematically: take each term from the first polynomial and multiply it by every term in the second. Write them all out, then combine like terms.

Q: How can I check if my expansion is correct?

Count your terms! $ (3y+x)(4+2x) $ should give you 4 terms initially (2×2), then combine any like terms. Also substitute simple values like x=1, y=1 to verify both sides equal the same number.

Q: What if I get confused with the signs?

Be extra careful with positive and negative signs! In this problem, all terms are positive, but always track signs carefully when multiplying. A positive times a positive gives positive.

Polynomial Multiplication with Distributive Property

Is equality correct?

$(3y+x)(4+2x)=2x^2+6xy+4x+12y$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Are the expressions equal?

00:05 Let's properly open parentheses and multiply each factor by each factor

00:23 Let's calculate the products

00:39 Let's compare the terms of the expressions, we'll see they're equal

00:46 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

Is equality correct?

$(3y+x)(4+2x)=2x^2+6xy+4x+12y$

Step-by-step solution

To determine if the equality $(3y+x)(4+2x)=2x^2+6xy+4x+12y$ is correct, we need to expand and simplify the left-hand side to see if it equals the right-hand side.

First, we expand $(3y+x)(4+2x)$ using the distributive property:

Multiply $3y$ by $4$ , giving $12y$ .
Multiply $3y$ by $2x$ , giving $6xy$ .
Multiply $x$ by $4$ , giving $4x$ .
Multiply $x$ by $2x$ , giving $2x^2$ .

Combining all these terms, the left-hand side expands to: $12y + 6xy + 4x + 2x^2$ .

Notice that this is precisely the same as the right-hand side: $2x^2 + 6xy + 4x + 12y$ .

Therefore, the equality $(3y+x)(4+2x) = 2x^2 + 6xy + 4x + 12y$ holds true.

Thus, the solution to the problem is: $(3y+x)(4+2x)=2x^2+6xy+4x+12y$ is correct, and the answer choice is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Distributive Property: Each term in first polynomial multiplies each term in second
FOIL Method: (3y+x)(4+2x) gives 12y + 6xy + 4x + 2x²
Verification: Rearrange expanded form to match given expression exactly ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply every term by every other term
Don't skip multiplication steps like only doing 3y×4 and x×2x = incomplete expansion! This misses crucial terms like 3y×2x and x×4. Always multiply each term in the first polynomial by every term in the second polynomial.

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

Do I have to multiply every term with every other term?

Yes, absolutely! When multiplying $(3y+x)(4+2x)$ , you need four multiplications: 3y×4, 3y×2x, x×4, and x×2x. Missing any step gives the wrong answer.

Why does the order of terms matter when checking equality?

The order doesn't matter in polynomial equality! $12y + 6xy + 4x + 2x^2$ equals $2x^2 + 6xy + 4x + 12y$ because addition is commutative.

What's the easiest way to expand two polynomials?

Use the distributive property systematically: take each term from the first polynomial and multiply it by every term in the second. Write them all out, then combine like terms.

How can I check if my expansion is correct?

Count your terms! $(3y+x)(4+2x)$ should give you 4 terms initially (2×2), then combine any like terms. Also substitute simple values like x=1, y=1 to verify both sides equal the same number.

What if I get confused with the signs?

Be extra careful with positive and negative signs! In this problem, all terms are positive, but always track signs carefully when multiplying. A positive times a positive gives positive.

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