Why when translating a binomial to trinomial does the sign switch

Question

If i have a homework problem that is (t - 3) (t² +3t + 9), I am factoring this, why does the sign change though? My teacher isn't really hapling me with this and I want to knoow why the sign changes and how to factor it. Thank you.

Comments

are you trying to multiply the 2 equations and then factor or are you trying to factor the largest equation

Answer 1

(t-3)(t²+3t+9)=t(t²+3t+9)-3(t²+3t+9)=

t³+3t²+9t-3t²-9t-27=t³-27 or t³-3³ (difference of two cubes).

(t+3)(t²-3t+9)=t(t²-3t+9)+3(t²-3t+9)=

t³-3t²+9t+3t²-9t+27=t³+27 or t³+3³ (sum of two cubes).

This shows how the difference and sum of two cubes factorises. Notice how the minus sign in either the first factor or the second factor causes terms to cancel out just leaving the cubes.

This "trick" applies to the sum and difference of other powers. For example, sixth power:

(x-y)(x⁵+x⁴y+x³y²+x²y³+xy⁴+y⁵)=

x⁶+x⁵y+x⁴y²+x³y³+x²y⁴+xy⁵-x⁵y-x⁴y²-x³y³-x²y⁴-xy⁵-y⁶=

x⁶-y⁶ (y can also be any number).

(x+y)(x⁵-x⁴y+x³y²-x²y³+xy⁴-y⁵)=

x⁶-x⁵y+x⁴y²-x³y³+x²y⁴-xy⁵+x⁵y-x⁴y²+x³y³-x²y⁴+xy⁵-y⁶=

x⁶-y⁶. Note how the alternating position of the minus sign causes all but two terms to cancel out.

Knowing this trick enables you to factor the sum or difference of the same integer powers (power must be greater than 2) of two different quantities. Although you can also factor x²-y², x²+y² does not factor into real factors.