Bellwork: Factor Each.
1. 5x2 + 22x + 8
2
2. 81x – 36
3. 7x2 – 27x – 4
4. 4x2 – 25
5. 20x2 – 7x – 6
6. 4x2 – 24x
Section 6.4
Factoring and Solving
Polynomials
Things to Know
Always check for GCF first!
Learn your perfect squares:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,
144, 169, 196, 225...
Learn your perfect cubes:
1, 8, 27, 64, 125, 216, 343, 512...
Sum or Difference of
Two Cubes
(a3 + b3) = (a + b)(a2 – ab + b2)
3
3
2
2
(a – b ) = (a – b)(a + ab + b )
SOAP: same, opposite, always plus
Examples
1. x3 + 8
3. x3 – 27
2
2
(x + 2)(x – 2x + 4)
(x – 3)(x + 3x + 9)
2. 8x3 – 1
4. 8x3 – 125y6
2
(2x – 1)(4x +2x+1)
(2x–5y2)(4x2+10xy2+25y4)
Factor each Binomial
5. 125 + x3
7. 81x4 – 16
2
(5 + x)(25 – 5x+x )
**Not a perfect cube**
(9x2 – 4)(9x2 + 4)
5
6. 16x – 250 x
2
2
(3x–2)(3x+2)(9x2+4)
3
2x (8x – 125)
2
2
2x (2x–5)(4x +10x+25)
4
8. 64a – 27a
a(64a3 – 27)
a(4a–3)(16a2+12a+9)
Factor each binomial
9. 25x4 – 36
11. 125x9 – 27y3
**Not a Perfect Cube**
(5x3–3y)(25x6+15x3y+9y2)
(5x2 – 6)(5x2+ 6)
8
12. 9x – 25y
3
10. 8x + 27
2
(2x +3)(4x – 6x+9)
6
**Not a Perfect Cube**
(3x4 – 5y3)(3x4 + 5y3)
Factor by Grouping:
4 Terms Only
3
2
1. x – 2x – 9x + 18
3
2
(x – 2x )(– 9x + 18)
2. bx2+2a + 2b + ax2
2
2
(bx +2a)(+ 2b + ax )
x (x – 2) –9(x – 2)
No GCF, so reorder.
2
(bx2+2b)(+ 2a + ax2)
2
(x – 2)(x – 9)
(x – 2)(x – 3)(x + 3)
2
2
b(x + 2) + a(2 + x )
2
(x + 2)(b + a)
Factor by Grouping:
4 Terms Only
3. 8x3 – 12x2 – 2x + 3
3
4. 2x3 – x2 + 2x – 1
2
(8x – 12x )(–2x + 3)
(2x – x )(+ 2x – 1)
4x2(2x – 3) –1(2x –3)
x2(2x – 1) + 1(2x–1)
(2x – 3)(4x2 – 1)
(2x – 1)(x2 + 1)
(2x–3)(2x–1)(2x + 1)
3
2
Factor by Grouping:
4 Terms Only
5. x2y2– 3x2– 4y2+12
2 2
2
2
6. 32x5–8x3 +4x2 – 1
5
3
2
(x y – 3x )(– 4y +12)
(32x –8x )(+4x – 1)
x2(y2 – 3) – 4(y2 – 3)
8x3(4x2–1)+1(4x2–1)
(y2 – 3)(x2 – 4)
(4x2 – 1)(8x3 + 1)
2
(y – 3)(x – 2)(x + 2)
(2x–1)(2x+1)(2x+1)(4x2–2x+1)
Factoring Polynomials:
3 Terms
1. x4 – 8x2 – 9
2
2
3. 5x4 – 2x2 – 3
(x – 9)(x + 1)
(5x2 + 3)(x2 – 1)
(x – 3)(x + 3)(x2 + 1)
(5x2 + 3)(x – 1)(x + 1)
4
2
2. 3x – 8x + 4
2
2
(3x – 2)(x - 2)
4. 8x4 + 3x2 – 5
(8x2 – 5)(x2 + 1)
Factor each.
1. 16x4 – 81
6
4
2. 2x – 6x – 20x
3
3. 8x – 343
4. x4 – 6x2 – 27
2
3
2
5. 3x – 7x –12x + 28
4
2
6. 3x – x - 4
Solving
To solve, you factor the problem, each
factor must be factored down to at most
a quadratic.
To solve, put each factor equal to zero.
Solve each.
1. 2x5 + 24x = 14x3
5
3
2x – 14x + 24x = 0
2x(x4 – 7x2 + 12) = 0
2
2
2x(x – 3)(x – 4) = 0
2
2
2x=0 x –3=0 x –4=0
X=0
x2=3
x2= 4
x=+√3 x = + 2
2. 2y5 – 18y = 0
4
2y(y – 9) = 0
2y(y2 – 3)(y2 + 3) = 0
2y=0
y2–3=0
y=0
y2=3
y=+√3
y2+3=0
y2=–3
y=+i√3
Solve each.
3. 3x4+3x3=6x2+6x
4
3
2
4. 3x7 = 81x4
7
4
3x +3x – 6x – 6x = 0
3x – 81x = 0
3x(x3+ x2– 2x – 2)= 0
3x4(x3 – 27) = 0
3x[x2(x+1) – 2(x + 1)] = 0
3x4(x–3)(x2+3x+9) = 0
3x(x +1)(x2 – 2) = 0
3x4= 0 x–3=0 x2+3x+9 = 0
3x=0
x+1=0 x2–2=0
x=0 x= –1 x2 = 2
x=+√2
x=0
x=3
Not real!
Solve each.
6. 4x4 – 8x2 = 5
3
5. 5x = 40
4x4 – 8x2 – 5 = 0
5x3 – 40 = 0
5(x3 – 8) = 0
(2x2 + 1)(2x2 – 5) = 0
5(x – 2)(x2 + 2x + 4)=0
2x2+1=0 2x2-5=0
5 = 0 x – 2 = 0 x2 + 2x + 4= 0
NS
x=2
2x2 = -1
2x2 = 5
2 imag solutions x2 = -1/2
x2 = 5/2
x =+i√2/2 x=+√10/2
Solve each.
1. x3 = 64
4
3. x3 + 3x2 = x + 3
2
2. x – 6x = 27
4
4. 16x – 25 = 0
Word Problems
●
The revenue in thousands of dollars for a small business
can be modeled by R = t3 – 8t2 + t + 82, where t is the
number of years since 1990. What was the revenue in
2002? In what year did the revenue reach $90,000?
Word Problems
A rectangular shipping container has a volume of
2500 cm3. The container is 4 times as wide as it
is deep, and 5cm taller than it is wide. What are
the dimensions of the container?
Word Problems
Between 1985 through 1995, the number of home computers, in thousands,
sold in Canada is estimated by this equation c(t) = 0.92(t3 + 8t2 + 40t + 400),
where t is the number of years since 1985. How many computers where
there in 1990? In what year did home computer sales reach 1.5 million?
Word Problems
A box with an open top is to be constructed from a
rectangular piece of cardboard by cutting out equal
squares of side x at each corner and then folding up the
sides If the dimensions of the cardboard are 12 by 20,
express the volume v of the box as a function of x. Tell
the domain of V(x). Find the height of the box if the
volume of the box is
Word Problems
A box with no top is to be made from an 8 inch by 6 inch piece of metal
by cutting identical squares from each corner and turning up the sides.
The volume of the box is modeled by the polynomial 4x -28x +48x.
Factor the polynomial completely. Then use the dimensions given on the
box and and show that its volume is equivalent to the factorization that
you obtain.
3
2
Word Problems
A catering company is designing a box. The volume box is to be 54 cubic
inches and the bottom of the box to be a square. Suppose the bottom of
the box has a width that is 3 inches smaller than the height x of the box.
write a polynomial equation of the box.