Question

How do you multiply polynomials `(7x^3 - 6x^2 + 4x-8) (9x^3 - 2x^2 - 3x +5)`?

Answer 1

Collect the coefficients of the product using a grid to find:

`(7x^3-6x^2+4x-8)(9x^3-2x^2-3x+5)`

`=63x^6-68x^5+27x^4-27x^3-26x^2+44x-40`

Explanation:

One way is to write out a grid of the coefficients and their products as follows:

Each reverse diagonal corresponds to choices of terms from each of the two original polynomials that result in the same power of `x` when multiplied.

Sum the reverse diagonals to get the coefficients of the product of the polynomials:

`(7x^3-6x^2+4x-8)(9x^3-2x^2-3x+5)`

`=color(red)(63)x^6+(color(green)(-54-14))x^5+(color(blue)(36+12-21))x^4+(color(purple)(-72-8+18+35))x^3+(color(orange)(16-12-30))x^2+(color(teal)(24+20))x+color(cyan)(-40)`

`=color(red)(63)x^6-color(green)(68)x^5+color(blue)(27)x^4-color(purple)(27)x^3-color(orange)(26)x^2+color(teal)(44)x-color(cyan)(40)`

Answer 2

`63x^6-68x^5+27x^4-27x^3-26x^2+44x-40`

Explanation:

#{: (" X ", " | ", 9x^3,-2x^2,-3x,+5), ("-----","-+-","-------","-------","-------","--------"), (7x^3," | ",color(orange)(63x^6),color(red)(-14x^5),color(blue)(-21x^4),color(green)(35x^3)), (-6x^2," | ",color(red)(-54x^5),color(blue)(12x^4),color(green)(18x^3),color(brown)(-30x^2)), (+4x," | ",color(blue)(36x^4),color(green)(-8x^3),color(brown)(-12x^2),color(purple)(20x)), (-8," | ",color(green)(-72x^3),color(brown)(16x^2),color(purple)(24x),color(cyan)(-40)) :}#

`=color(orange)(63x^6) color(red)(-14x^5-54x^5)color(blue)(+27x^4)color(green)(+35x^3+18x^3-8x^3-72x^3color(brown)(-30x^2-12x^2+16x^2)color(purple)(+20x+24x)color(cyan)(-40)`

`=color(orange)(63x^6) color(red)(-68x^5)color(blue)(+27x^4)color(green)(-27x^3color(brown)(-26x^2)color(purple)(+44x)color(cyan)(-40)`