Question

How do you FOIL with 3 terms?

`(8r^2 + 4r + 6)(3r^2 − 7r + 1)`

How do I foil when I have 3 terms?
Help and steps are much appreciated :)

Answer 1

`(8r^2+4r+6)(3r^2-7r+1)=24r^4-44r^3-2r^2-38r+6`

Explanation:

FOIL is a mnemonic to help enumerate all individual products of terms when multiplying two binomials. It captures the result of applying the distributive property of multiplication over addition three times:

`(a+b)(c+d) = a(c+d)+b(c+d)`

`color(white)((a+b)(c+d)) = overbrace(ac)^"First"+overbrace(ad)^"Outside"+overbrace(bc)^"Inside"+overbrace(bd)^"Last"`

FOIL is not applicable to trinomials, but distributivity is.

So we could solve the given problem by:

`(8r^2+4r+6)(3r^2-7r+1)`

`=8r^2(3r^2-7r+1)+4r(3r^2-7r+1)+6(3r^2-7r+1)`

`=(24r^4-56r^3+8r^2)+(12r^3-28r^2+4r)+(18r^2-42r+6)`

`=24r^4-56r^3+12r^3+8r^2-28r^2+18r^2+4r-42r+6`

`=24r^4+(-56+12)r^3+(8-28+18)r^2+(4-42)r+6`

`=24r^4-44r^3-2r^2-38r+6`

Alternatively, we can write the coefficients of the `9` individual products of pairs of terms in a table and sum the reverse diagonals, to find the coefficients of the product like this:

`underline(color(white)(+)color(white)(00) \ " |" color(white)(+)color(white)(0)8 \ color(white)(+)color(white)(0)4 \ color(white)(+)color(white)(0)6)`
`color(white)(+)color(white)(0)3 \ " |" color(white)(+)color(red)(24) \ color(white)(+)color(orange)(12) \ color(white)(+)color(green)(18)`
`color(black)(-)color(white)(0)7 \ " |" color(orange)(-)color(orange)(56) \ color(green)(-)color(green)(28) \ color(blue)(-)color(blue)(42)`
`color(white)(+)color(white)(0)1 \ " |" color(white)(+)color(white)(0)color(green)(8) \ color(white)(+)color(white)(0)color(blue)(4) \ color(white)(+)color(white)(0)color(purple)(6)`

Hence:

`(8r^2+4r+6)(3r^2-7r+1)`

`=color(red)(24)r^4+(color(orange)(-56+12))r^3+(color(green)(8-28+18))r^2+(color(blue)(4-42))r+color(purple)(6)`

`=24r^4-44r^3-2r^2-38r+6`

Alternatively, we could examine the given product of trinomials and think about each power of `r` in descending order, summing all the ways that a term in the first trinomial multiplied by a term in the second can give rise to that power. With practice, that should allow us to write the answer directly.

`(8r^2+4r+6)(3r^2-7r+1)=24r^4-44r^3-2r^2-38r+6`