1. Chapter 6 Section 4: Factoring and Solving Polynomials Equations

2. 1. Factor out any common monomials
3 𝑥 2 −12
=3 (𝑥 2 −4)
𝑥 3 −2 𝑥 4 +x
=𝑥(𝑥 2 −2 𝑥 3 +1)
6𝑥 7 −4 𝑥 2
=2 𝑥 2 (3𝑥 5 −2)

3. SPECIAL FACTORING PATTERNS
NAME
 Difference of
Two Squares
Perfect Square
Trinomial
a2 + 2ab + b2
= (a + b)2
PATTERN
x2 – y2
= (x + y)(x – y)
x2 + 12x + 36
= (x + 6)2
EXAMPLE
x2 – 9
= (x + 3)(x– 3)

4. 2. Look for special patterns
= (x + 4)(x– 4)
x2 – 16
x2 + 14x + 49
= (x + 7)2
There are other special patterns that are also worth remembering

5. SPECIAL FACTORING PATTERNS
NAME
 Difference of
Two cubes
Sum of two cubes
x3 + y3 =
(x + y)(x2 –xy+ y2)
PATTERN
x3 - y3 =
(x - y)(x2 +xy+ y2)
x3 + 8
= (x + 2)(x2 -2x+ 4)
EXAMPLE
8x3 – 1
(2x - 1)(4x2 +2x+ 1)

6. 3.Factoring by grouping 1. Begin by factoring out the GCF.
1. 5x3+2x2-40x-16
None
2. Arrange the four terms so that the first two terms and the last two terms have common factors.
2. (5x3+2x2)+(-40x-16)
3. If the coefficient of the third term is negative, factor out a negative coefficient from the last two terms.
3. (5x3+2x2)-(40x+16)
4. Use the reverse of the distributive property to factor each group of two terms.
4. x2(5x+2)-8(5x+2)
5. Now factor the GCF from the result of step 4 as done in the previous section.
5. (5x+2)(x2-8)

7. Factoring using quadratics
x4 + 3x2 -4
The following steps can be used to solve equations that are quadratic in form:
1. Let u equal a function of the original variable (normally the middle term)
2. Substitute u into the original equation so that it is in the form au2 + bu + c
3. Factor the quadratic equation using the methods learned earlier
4. Replace u with the expression of the original variable.
5. Factor again if necessary.
1. u=x2
2. u2+3u-4
3. (u+4)(u-1)
4. (x2+4)(x2-1)
5. (x2+4)(x-1)(x+1)

8. Solving Polynomials Remember
Finding zeros, solutions, and roots are different ways of saying the same thing.
So…
After you factor the polynomial, set it equal to 0.
Then solve the polynomial.

9. Find the real-number solutions
x4 + 3x2 -4=0
(x2+4)(x-1)(x+1)=0
(x2+4=0, not real
(x-1)=0 x=1
(x+1)=0 x=-1