Week 5: Factoring Polynomial Expressions

Published by on March 4, 2018 | Leave a response

*Trinomials general form: Ax^2 +Bx +C

**Factoring process:
– Step 1: Remove the common factor to get integer coefficients. Then remove another common factor to get smallest integer coefficients possible.
– Step 2: Think of two numbers that have their sum equals to B, and their product equals to AC
– Step 3: Use the two numbers as the coefficients of x, and replace Bx with the sum of the two new terms.
– Step 4: Remove a common factor from the first pair of terms, and another from a second pair of terms.
– Result: Each product has a common binomial factor used to complete the factoring process.

***Some polymial expressions contain functions of variable. Treat the function as a variable by replacing the function with another variable (z). After finishing factoring, substitute z with the old function then simplify to get the final result.

**Example:
1) x^2 -\frac{17}{3}x -2
=\frac{1}{3}(3x^2 -17x -6)
=\frac{1}{3}(3x^2 -18x +x -6) (-18 +1 =17 and 1(-18) =3(-6) =-18)
=\frac{1}{3}(3x(x-6) +(x-6))
=\frac{1}{3}(x-6)(3x+1)

2) 2(x-6)^2 +10(x-6) -48
Replace (x-6) with z: 2z^2 +10z -48
=2(z^2 +5z -24)
=2(z^2 +8z -3z -24) (8 +(-3) =5 and 8(-3) =-24)
=2(z(z+8) -3(z+8))
=2(z-3)(z+8)
=2(x-6-3)(x-6+8)
=2(x-9)(x+2)

*Difference of squares general form: x^2 - y^2

**Factoring process:
– Step 1: Remove the common factor
– Step 2: Rewrite the expression under the general form and use the fuormular to factor: x^2 - y^2 = (x-y)(x+y)

**Example:
1) (3x+4)^2 -(2y-1)^2
=((3x+4) +(2y-1))((3x+4) -(2y-1))
=(3x +4 +2y -1)(3x +4 -2y +1)
=(3x +2y +3)(3x -2y +5)

2) 27(2x-3)^2 -75(y-4)^2
=3( 9(2x-3)^2 -25(y-4)^2)
=3( (3(2x-3))^2 -(5(y-4))^2)
=3(3(2x-3) +5(y-4))(3(2x-3) -5(y-4))
=3(6x -9 +5y -20)(6x -9 -5y +20)
=3(6x +5y -29)(6x -5y +11)

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