Polynomial identities

Three quick questions from earlier lessons.

Without a calculator: $51^2 - 49^2$ . Which shortcut helps?

Identities are true for every value of the variables.

a^2 - b^2 = (a + b)(a - b)

;

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

;

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

;

(1 - r)(1 + r + \dots + r^{n-1}) = 1 - r^n

Factor $x^3 + 27$ .

a = x

b = 3

27 = 3³.

(x + 3)(x^2 - 3x + 9)

Sum of cubes.

The middle terms cancel

Check one term: x · 9 + 3 · (−3x) = 0x.

Your turn: $1 + 3 + 9 + 27$ with $r = 3$ , $n = 4$ : $(1 - 3^4)/(1 - 3)$ .

40

(1 − 81)/(−2).

Use $(1 - r)(1 + r + r^2 + \dots + r^{n-1}) = 1 - r^n$ to compute $1 + 2 + 4 + \dots + 2^{ 4 }$ .
Use $(1 - r)(1 + r + r^2 + \dots + r^{n-1}) = 1 - r^n$ to compute $1 + 2 + 4 + \dots + 2^{ 9 }$ .

Three questions to check you have got it.

Polynomial identities

$a^2 - b^2$ , $a^3 \pm b^3$ , and the geometric-sum identity.

Watch out: the sign pattern in the cube identities.

End of the lesson