Multiplication of Polynomials

Question

So, I'm reading through this old math book (title doesn't matter).

I'm in the section that talks about multiplying polynomials. I've read it and completely understand it. I did all the exercises and pretty much got all of them correct.

For extra practice though, I decided to write a few of my own problems and solve them.

Instantly, the technique that the book teaches just uberly fails. Here are some examples of questions I made up:

$$(3x^2 + 4xy + 5y^2 + 4x^2y)(3x - 4y^3 + 7x^2)$$

$$(3xy + 4y^2 - 7x)(5x + 2y)$$

I just can't see why these aren't solvable and

$$(2a - 3b)(3a^2 - 2ab + 4b^2)$$

$$(2x^2 - 3xy + y^2)(3x^2 - xy + 2y^2)$$

...the above ones are. What's the difference?

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For the record, the book teaches that the problem should be solved by distributing the polynomials and adding the like-terms. The problem I'm having with my expressions is that I'm not getting like-terms.

Answer 1

For example, in your first case you should get $$(3x^2 + 4xy + 5y^2 + 4x^2y)(3x - 4y^3+7x^2) = -16\,{x}^{2}{y}^{4}+28\,{x}^{4}y-12\,{x}^{2}{y}^{3}-16\,x{y}^{4}-20\,{ y}^{5}+21\,{x}^{4}+40\,{x}^{3}y+35\,{x}^{2}{y}^{2}+9\,{x}^{3}+12\,{x}^ {2}y+15\,x{y}^{2} $$ There are $11$ terms on the right side, while the two factors had $4$ and $3$. So there were just two "like terms" that combined into one. Well, that's life: sometimes there may be more, sometimes less.

Answer 2

Like terms are those which have same variables no matter coefficients are similar or different.

e.g. $4 x^{2}$ and $ 6 x^{2},$

$2x^3yz$ and $ -6x^3yz$ are like terms.

$(3xy+4y^2−7x)(5x+2y)$

$=3xy(5x+2y)+4y^2(5x+2y)$

$= 15x^2y +6xy^2 + 20xy^2 + 8y^3 - 35x^2 -14xy$

here, $6xy^2$ and $20xy^2$ are like terms

$= 15x^2y + 26xy^2 + 8y^3 - 35x^2 -14xy $

Apply this method using different expression.