Factoring Reviewed

Factoring - Reviewed

Monomials, Binomials, Trinomials

& More…

A Quadratic Activity

Scratching Your Head Over Factoring Quadratics?

Then you must get back to basics

Quadratic Activity A to Z

?

Back to Basics With Quadratics

If you listen attentively And keep eyes and hands busy

Factoring Quadratics can be as easy

as 123

+ ÷ - x

Linear

Circular

Parabola

•  Everything’s not straight. •  The power of 1; that’s me

•  Y = mX + b

•  Some things are round •  The power of 2; add you

•  X2 + Y2 = r2

•  Other things are turned upside down! •  The power of 1 and 2; that’s me and you

•  Y = a X2 + b X + c -

1. Greatest Common Factor

In order to factor completely You must consider the GCF

Factoring out the GCF is like studying a family with children. • Let m2w + mw2 represent the family unit, where m represents the father, and let w represents the mother.

• What does the family unit have in common? • You must factor to find out: mw(m + w) • So mw represents the child, the greatest common factor in the family.

• If there is more than one child, the family could look like this: m3w2 + m2w3 • Factor out all the children: (mw)(mw)(m + w) = m2w2(m + w)

2. Factoring and Special Products

1. Sometimes Alone

2. Sometimes together

- Always work out the difference between each other

(x + y)2 = (x + y)(x + y) (x – y)2 = (x - y)(x – y)

(x + y)(x – y)

x2 – y2

Example: (2x2 - 12x + 18) 2(x2 - 6x + 9) – That took care of the GCF. ? Is the first term a perfect square? ? Is the third term a perfect square? ? Is the 3rd term ½ the middle coefficient squared? (6/2 = 3, and 32 = 9) ? If the answer to all three is yes, then: (2x2 - 12x + 18) factors completely as 2(x – 3)(x – 3)

2(x – 3)2 ; and that’s a perfect square.

Special Products Cont’d

1. Sometimes Alone - Factoring Perfect Squares

Don’t recoil, just remember how to F.O.I.L. (x2 – xy + xy - y2) You may be asking, “Why should I care?” ‘Cause when you bring them together, the middles disappear!

(x2 – xy + xy - y2) = x2 - y2 Let’s Teach By Example: Example 1: 4x2 – 25 - 4 is perfect - x2 is perfect - 25 is perfect. Now let’s un-F.O.I.L. it.


2. Sometimes Together - Factoring Binomials (x + y)(x – y)

To Factor Difference of Squares – Mix the signs

and take the square root. √x2 = x; √4 = 2; √25 = 5 4x2 – 25 = (2x – 5)(2x + 5)

It’s not always clear that the square is there

Example 2: -27x3 + 75x •  Think again bout factoring completely; •  You must remember the GCF

-3x(9x2- 25)

-27x3 + 75x = -3x(3x – 5)(3x + 5)


1.  There is no GCF 2.  The factors of 6 are (1, 6), (-1, -6) and (2, 3), (-2.

-3) 3.  None of the sums or differences of the pairs of

factors of 6 add up to the middle term, -3. 4.  Conclusion: This trinomial cannot be factored

x2 -3x +6 is prime

1. Prime x2 -3x +6

3. Factoring Trinomials

1.  There is no GCF 2.  The factors of 6 are (1, 6), (-1, -6) and (2, 3), (-2,

-3) 3.  The pair (-2, -3) sum to -5, the middle term. 4.  Conclusion: This trinomial is factorable 5.  Write x as the first term in the pair of parentheses

(x )(x ) 6.  Write each factors – with its sign – as the second

term of each parenthesis (x - 2)(x - 3) The trinomial with first coefficient ‘1’ is done!

Factoring Trinomials cont’d

x2 -5x +6 2. First coefficient is 1 - Factorable

1.  Split the middle term 1.  3p2 + 6p + 3p + 6 : Notice that the sum of the

split term equals the original middle term. 2.  Notice that the product of the split term (6)(3) =

18 = (3)(6), the product of the first and last term. 2.  Group: Create binomials with the first two terms and

the last two terms (3p2 + 6p) + (3p + 6)

- If it’s 3 then split it - When it’s 4 then group it

1. Trinomials – First Coefficient Not ‘1’ 3p2 + 9p + 6

.

4. Factoring by Grouping

3.  Factor out the GCF – 3p(p + 2) + 3(p + 2) 4.  Then factor out the GCF again

(p + 2)(3p + 3) Notice that the GCF in this case is (p + 2)

Factoring by Grouping

1. Trinomials – First Coefficient Not ‘1’ Cont’d

When split in 4, group and factor. 1.  (7q3 + q2)+ (14q + 2) 2.  Factor Out the GCF – q2(7q + 1)+ 2(7q +1) 3.  And Factor out the GCF again

(7q + 1)(q2 + 2)

2. Four Terms – Group them – 7q3 + q2 + 14q + 2

1.  Look at c3 + d3 – The first and second terms are cubes

2.  Take the cube root of both terms: cube root of c3, and d3 is c, and d respectively.

3.  Then write the product of a binomial and a trinomial

You Can’t get out of the cube without factoring.

5.Factoring Cubes

Factoring Cubes – Is All About Products 1. Sum of Cubes

The Sum and Difference of Cubes

1. Sum of Cubes Cont’d 1.  Write the binomial (c + d) 2.  Now create a trinomial like this:

1.  Square the first term of the binomial c2 – that’s a product - and place it as the first term of a trinomial (c2 )

2.  Square the second term of the binomial, d2 - that’s a product - and place it as the third term of the trinomial (c2 d2)

3.  Find the product of the first and second terms of the binomial and place it as the middle term of the trinomial. (c2 cd d2).



1. Sum of Cubes Cont’d 1.  Now What about the signs?

1.  The sign of the binomial is always the same as the sign of the cube. (c + d)

2.  The first sign in the trinomial is always the opposite of the sign of the binomial (c2 – cd d2).

3.  The second sign in the trinomial is always positive (c2 – cd + d2).



2. Difference of Cubes 1.  Let’s show this one by Example 2.  8q3 – 27

1.  The cube root of 8 is 2 2.  The cube root of q3 is q 3.  The cube root of 27 is 3

3.  Using the same general rules as for the sum of squares 1.  Rewrite 8q3 – 27 as (2q)3 – 33. 2.  8q3 – 27 = (2q – 3)((2q)2 – (2q)(3) + 32) (2q – 3)(4q2 – 6q + 9)

The Sum and Differene of Cubes

Education

Factoring Reviewed