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I can't figure out how to multiply these polynominals $(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$

I tried multiplying like this $(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)$ $3x^4-7x^3+(5x^2)(2x^2)(-6x^2)+(5x)(7x)(-4x)+(8)(9)$ $3x^4-7x^3-60x^2-140x+72$

It says the answer is $-12x^6-19x^5-14x^4-68x^3+28x^2+69x+72$ but how did they get it?

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    A first step might be to arrange each of the polynomials in descending powers of $x$. That will help keep things more organized.2012-12-08

2 Answers 2

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You have to multiply every term by every other term. A good way to make sure you don't miss any is to use a table.

First, combine like terms within each group of parentheses: $(5x^2+3x^4-7x^3+5x+8)(2x^2-4x+9-6x^2+7x)\\ =(3x^4-7x^3+5x^2+5x+8)(-4x^2+3x+9)$

Then form a table and multiply each term by multiplying the coefficients and adding the exponents: $ \begin{array}{c|cc} \text{} & 3x^4 & -7x^3 & 5x^2 & 5x & 8 \\ \hline -4x^2 & -12x^6 & -28x^5 & -20x^4 & -20x^3 & -32x^2 \\ 3x & 9x^5 & -21x^4 & 15x^3 & 15x^2 & 24x \\ 9 & 27x^4 & -63x^3 & 45x^2 & 45x & 72 \\ \end{array} $

Now take the new polynomial from the table and combine like terms: $-12x^6-28x^5-20x^4-20x^3-32x^2+9x^5-21x^4+15x^3\\ +15x^2+24x+27x^4-63x^3+45x^2+45x+72\\$$ =-12x^6+(9-28)x^5+(27-21-20)x^4+(-63+15-20)x^3\\+(45+15-32)x^2+(45+24)x+72\\$$ =-12x^6-19x^5-14x^4-68x^3+28x^2+69x+72$

This method will also work with negative and non-integer exponents, as it is not restricted to polynomials.

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    @AustinMohr While it is not a polynomial, I did mention that "this method will also work with negative and non-integer exponents," as the table method is not restricted to polynomials. Perhaps I should have stated that more clearly in the answer.2012-12-08
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To multiply two polynomials, distribute each term in the left set of parentheses over the entire collection of terms in the right set of parentheses. Your example would start out like:

$ \begin{align*} & (\color{red}{5x^2+3x^4-7x^3+5x+8})(2x^2-4x+9-6x^2+7x)\\ = & \color{red}{5x^2}(2x^2-4x+9-6x^2+7x)\\ & \color{red}{+ 3x^4}(2x^2-4x+9-6x^2+7x)\\ & \color{red}{- 7x^3}(2x^2-4x+9-6x^2+7x)\\ & \color{red}{+ 5x}(2x^2-4x+9-6x^2+7x)\\ & \color{red}{+ 8}(2x^2-4x+9-6x^2+7x). \end{align*} $

Can you take it from here?

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    I copied your expression exactly, but notice the righthand polynomial can be further simplified to $-4x^2 + 3x + 9$.2012-12-08