Replace each boxed question mark with a polynomial that results i...

Question

Replace each boxed question mark with a polynomial that results in the given product.

2x³y² · ? = 12x⁵y⁴

Accepted Answer

Hey, everyone in this problem, we're asked to solve the following equation for a, the equation we're given is seven X to the exponent five. Why? To the exponent eight multiplied by A is equal to 35 x to the exponent nine Y to the exponent 12. We're given four answer choices. Option A five X squared Y squared B X to the exponent 45 Y to the exponent C5 x to the exponent 4 y to the exponent four N D seven X to the exponent 14 Y to the exponent 20. Now let's rewrite the equation we were given, we have seven X to the exponent five Y to the exponent eight multiplied by A is equal to 35 X to the exponent 9 y to the exponent 12. Now, this equation might look a little messy but we don't wanna get overwhelmed. We're gonna solve this just like we would an equation with more simple exponents or just numbers. OK. We wanna solve for a, that means we wanna isolate for A. So we need to get rid of this seven X to the exponent five Y to the exponent eight, that is multiplying the A. Now, in order to get rid of this term that's multiplied, we need to do the opposite, we need to divide. And so we're gonna have a, A is equal to 35 x to the exponent nine Y to the exponent 12 divided by seven X to the exponent five Y to the exponent eight. So now we've isolated for A and the problem becomes a matter of simplifying this expression. On the right hand side, we have a is equal to, let's deal with the coefficient first. So the number we have 35 divided by seven, we know that 35 divided by 7 is 5. So we're gonna have a coefficient of five. Now we have X to the exponent nine divided by X to the exponent five. Recall when we have two things that are being divided that have the same base, the exponents subtract. And what do I mean? Well, if we have B to the exponent M divided by B to the exponent N, this is equivalent to writing B to the exponent M minus N. So we keep the same base. We take the exponent from the numerator and subtract the exponent from the denominator. If we apply this to our problem, we have X to the exponent nine, the exponent in the numerator -5, the exponent in the denominator. And we can apply the same rule for Y. So we have Y to the exponent 12 from the numerator -8 from the denominator. OK. Again, we've simplified quite a bit more. Now, we can go ahead and do the subtraction in these exponents to simplify those exponents to a single number. Can we get A is equal to five X to the exponent four Y to the exponent four. And that's the solution for A, that's been simplified as far as we can simplify if we compare this to our answer choices, we found that A is equal to five X to the exponent four, Y to the exponent four, which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

Replace each boxed question mark with a polynomial that results in the given product. 2x³y² · ? = 12x⁵y⁴

Key Concepts

Polynomials

Multiplication of Polynomials

Degree of a Polynomial

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