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

Oct 13, 2015

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

$\left(7 {x}^{3} - 6 {x}^{2} + 4 x - 8\right) \left(9 {x}^{3} - 2 {x}^{2} - 3 x + 5\right)$

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

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:

$\left(7 {x}^{3} - 6 {x}^{2} + 4 x - 8\right) \left(9 {x}^{3} - 2 {x}^{2} - 3 x + 5\right)$

$= \textcolor{red}{63} {x}^{6} + \left(\textcolor{g r e e n}{- 54 - 14}\right) {x}^{5} + \left(\textcolor{b l u e}{36 + 12 - 21}\right) {x}^{4} + \left(\textcolor{p u r p \le}{- 72 - 8 + 18 + 35}\right) {x}^{3} + \left(\textcolor{\mathmr{and} a n \ge}{16 - 12 - 30}\right) {x}^{2} + \left(\textcolor{t e a l}{24 + 20}\right) x + \textcolor{c y a n}{- 40}$

$= \textcolor{red}{63} {x}^{6} - \textcolor{g r e e n}{68} {x}^{5} + \textcolor{b l u e}{27} {x}^{4} - \textcolor{p u r p \le}{27} {x}^{3} - \textcolor{\mathmr{and} a n \ge}{26} {x}^{2} + \textcolor{t e a l}{44} x - \textcolor{c y a n}{40}$

Oct 13, 2015

$63 {x}^{6} - 68 {x}^{5} + 27 {x}^{4} - 27 {x}^{3} - 26 {x}^{2} + 44 x - 40$
{: (" 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)