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

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

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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