Investigation: Basic Factoring
Factoring is undoing multi plication, almost like division .
Part 1: Factoring with Numbers
Step 1: Factor by making a factor tree:
a. 108 b. 200 c. 36
Step 2: What is the greatest common factor between
a. 108 and 200 b. 200 and 36 c. 108 and 36
Part 2: How to factor out the GCF.
Step 1: Simplify the following. Put your answer in standard form.
a. 8(3) =
b. 4(3 + 2) =
c. 2x(5 – x) =
d. 3t2(4t3 + 7) =
e. -(n + 3) =
f. (9m3 – 8m4 + 6m)(-7m2) =
Step 2: Complete each statement by factoring out the greatest common factor.
a. 4x2 + 20x – 12 = _____ · (x2 + 5x – 3)
b. 9n2 – 24n = 3n( ________________)
c. 9x2 + 3x – 18 = __________ ·(_______________________________)
d. 7p2 + 21 = ___________________________________
e. 4w3 + 2w2 =
f. 18x2y3z – 24x2yz + 6xy2z + 8xyz =
g**. 3x(x + 3) + 2(x + 3) =
Step 3: Now simplify each of the following. Put your answer in standard form. What do you notice about the relationship between the middle term and the last term?
a. (x – 5)(x + 3) =
b. (x - 4)(x – 1) =
c. (x + 2)(x + 7) =
d. (x + 4)(x – 9) =
Step 4: Now complete each statement to factor. Check your answers by multiplying the factors.
a. x2 + 8x + 7 = (x + 1)(x + ___)
b. x2 + 6x + 8 = (x + 2)(___ + ____)
c. x2 + 12x + 32 = (_____________)(x + 4)
d. x2 + 14x + 40 = (________________)(________________)
e. x2 – 17x + 72 = (x – 9)(_____________________)
f. x2 – 6x + 8 = (__________________)(x – 4)
g. x2 – 7x + 12 = (_____________________)(___________________)
h. x2 – 11x + 24 =
i. x2 – x – 12 = (x – 4)(____________________)
j. x2 – 14x – 32 = (x + 2)(_______________________)
k. x2 + 9x – 10 = (____________________)(___________________________)
l. x2 + 4x – 5 =
m. 3x2 – 9x – 84 =
Step 5: Now simplify these. Put your answer in standard form.
a. (2x – 7)2
b. (4x + 3)(4x – 3)
c. (x – 9)(x + 9)
d. (5x + 4)2
Step 6: Factor each of the following.
a. 9x2 – 42x + 49 = (_____________)(________________)
b. 4x2 + 12x + 9 =
c. 64x2 – 16x + 1 =
d. 25x2 + 90x + 81
e. x2 – 64
f. 4a2 - 49
Step 7: In general when factoring an expression of the form
x2 + bx + c, you want to try to find 2 numbers that (multiply/add) ________ to b and (multiply/add) ________ to c.
a2 + 2ab + b2 = ____________________ is called a perfect square trinomial.
a2 – b2 = __________________ is called a difference of squares.
Step 8: Put it all together.
Factor.
a. 15x3y5z2 – 30x2y7z + 20x2y3z5 – 5x2y3z
b. 3x(x +2) + 7(x + 2)
c. 3x2+ 12x - 63
d. 100x2 - 49
e. 9x2 + 30xy + 25y2
f. x2 – 12x - 32