General formula for multiplying two degree-two polynomials together?

Question

I am trying to figure out a general formula for getting the product of two degree-2 polynomials.

For example, I have $ax^2+bx+c$ and want to multiply it by $dx^2+ex+f$ where all variables are constants except for $x$. What would be the product of these two polynomials?

I tried computing the general formula, but I don't believe I got the correct answer. Thanks for your help!

What I got was $adx^4 + (ae+bd)x^3 + (af+be+cd)x^2 + (ce+bf)x + cf$.

score 2 · Answer 1 · answered Feb 01 '14 at 06:15

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$$(a_{_2}x^2+a_{_1}x+a_{_0})\cdot(b_{_2}x^2+b_{_1}x+b_{_0})=\sum_{k=0}^4\sum a_i\cdot b_{k-i}\cdot x^k$$

answered Feb 01 '14 at 06:15

Lucian

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With the added mention that both i and $k-i$ have to be in between $0$ and $2$. – Lucian Feb 01 '14 at 06:16

André Nicolas · Answer 2 · 2014-02-01T06:05:41.893

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We have $$(ax^2+bx+c)(dx^2+ex+f)=ax^2(dx^2+ex+f)+bx(dx^2+ex+f)+c(dx^2+ex+f).$$ Do the multiplications mentioned on the right. We get $$(adx^4+aex^3+afx^2)+(bdx^3+bex^2+bfx)+(cdx^2+cex+cf).$$ Now gather like powers of $x$ together. We get $$adx^4+(ae+bd)x^3+(af+be+cd)x^2+(bf+ce)x+cf.$$

Added: The answer you added is correct.

edited Feb 01 '14 at 06:05

answered Feb 01 '14 at 06:00

André Nicolas

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Oh okay, so I was correct then. Interesting... Ok. Thank you! And do you know if this should work for any constants plugged into a, b, c, d, e, or f? – Logan Feb 01 '14 at 06:01
When I was typing you had not yet written what you got. And you were not off at all, I had a typo. – André Nicolas Feb 01 '14 at 06:03
Note that the answer by Lucian is better, since it readily generalizes. – André Nicolas Feb 01 '14 at 06:23

General formula for multiplying two degree-two polynomials together?

2 Answers2