MATH 31 LESSONS

PreCalculus

1. Simplifying and Factoring Polynomials

A. Simplifying Polynomials

When you simplify a polynomial,

you are removing the brackets.

e.g.

(2x - 3) (4x + 1) = 8x2 - 10x - 3

Also, you are reducing a polynomial to the smallest

number of terms.

1. Adding and Subtracting Polynomials

You can add or subtract monomials

only with like terms.

e.g.

5x + 7x = 12x

11y2 - 7y2 = 4y2

6ab3 + 11ab3 = 17ab3

If they are not like terms,

then you cannot add them.

e.g.

2x + 3y

5y2 - 8y3

12xy2 + 8x2y

Ex. 1 Simplify 2x - 11y + 7x + 3y + 5x

Try this example on your own first.Then, check out the solution.

2x - 11y + 7x + 3y + 5xIdentify the like terms

2x - 11y + 7x + 3y + 5x

= 2x + 7x + 5x - 11y + 7yCollect the like terms

2x - 11y + 7x + 3y + 5x

= 2x + 7x + 5x - 11y + 3y

= 14x - 8y

2. Multiplying Polynomials

Monomial Monomial

Consider

5a2b3 10ab4 =

5a2b3 10ab4 = (5 10) (a2 a) (b3 b4)

Multiply numbers and like variables separately

5a2b3 10ab4 = (5 10) (a2 a) (b3 b4)

= 50 a3 b7

Monomial Polynomial

Consider

5x (6x - 7) =

5x (6x - 7) = 5x (6x) - 5x (7)

Multiply the monomial to each term of the polynomial

5x (6x - 7) = 5x (6x) - 5x (7)

= 30x2 - 35x

Binomial Binomial

Consider

(2x - 3) (4x + 1) =

(2x - 3) (4x + 1) = 2x (4x)

Use FOIL: First

(2x - 3) (4x + 1) = 2x (4x) + 2x (1)

Use FOIL: First

Outside

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) - 3 (4x)

Use FOIL: First

OutsideInside

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) - 3 (4x) - 3 (1)

Use FOIL: First

OutsideInsideLast

(2x - 3) (4x + 1) = 2x (4x) + 2x (1) - 3 (4x) - 3 (1)

= 8x2 + 2x - 12x - 3

= 8x2 - 10x - 3

Polynomial Polynomial

Consider

(x + 2y) (5x - 3y + 6) =

(x + 2y) (5x - 3y + 6) = x (5x) - x (3y) + x (6)

Multiply the first term to the entire polynomial

(x + 2y) (5x - 3y + 6) = x (5x) - x (3y) + x (6)

+ 2y (5x) - 2y (3y) + 2y (6)

Then, multiply the second term to the entire polynomial

(x + 2y) (5x - 3y + 6) = x (5x) - x (3y) + x (6)

+ 2y (5x) - 2y (3y) + 2y (6)

= 5x2 - 3xy + 6x + 10xy - 6y2 + 12y

= 5x2 + 6x + 7xy - 6y2 + 12y

Ex. 2 Simplify 2 (3a + 4) (5a - 6) - (2a - 7)2

Try this example on your own first.Then, check out the solution.

2 (3a + 4) (5a - 6) - (2a - 7)2

= 2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7)

If it is a perfect square, then you should write both binomials. Then, you will remember to FOIL.

Notice:

(2a - 7)2 (2a)2 - (7)2

2 (3a + 4) (5a - 6) - (2a - 7)2

= 2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7)

= 2 (15a2 - 18a + 20a - 24) - (4a2 - 14a - 14a + 49)

Be certain to show the brackets around the entire product

2 (3a + 4) (5a - 6) - (2a - 7)2

= 2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7)

= 2 (15a2 - 18a + 20a - 24) - (4a2 - 14a - 14a + 49)

= 2 (15a2 + 2a - 24) - (4a2 - 28a + 49)

2 (3a + 4) (5a - 6) - (2a - 7)2

= 2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7)

= 2 (15a2 - 18a + 20a - 24) - (4a2 - 14a - 14a + 49)

= 2 (15a2 + 2a - 24) - (4a2 - 28a + 49)

= 30a2 + 4a - 48 - 4a2 + 28a - 49

Distribute the negative to all terms

2 (3a + 4) (5a - 6) - (2a - 7)2

= 2 (3a + 4) (5a - 6) - (2a - 7) (2a - 7)

= 2 (15a2 - 18a + 20a - 24) - (4a2 - 14a - 14a + 49)

= 2 (15a2 + 2a - 24) - (4a2 - 28a + 49)

= 30a2 + 4a - 48 - 4a2 + 28a - 49

= 26a2 + 32a - 97 Add like terms

B. Factoring Polynomials

When you factor a polynomial,

you are adding brackets.

e.g.

8x2 - 10x - 3 = (2x - 3) (4x + 1)

You are making a polynomial into a product.

1. Greatest Common Factor (GCF)

The GCF is:

the largest number that divides evenly into

the coefficients

the smallest power of each variable

Taking out the GCF is usually the first step of factoring.

e.g.

Factor 12 x3 y4 + 18 x8 y2

12 x3 y4 + 18 x8 y2

= 6 x3 y2 (

The largest number that divides into 12 and 18 evenly

The smallest power of each variable

12 x3 y4 + 18 x8 y2

= 6 x3 y2 ( 2 x3-3 y4-2 + 3x8-3 y2-2 )

When you factor (divide), you subtract the exponents

12 x3 y4 + 18 x8 y2

= 6 x3 y2 ( 2 x3-3 y4-2 + 3x8-3 y2-2 )

= 6 x3 y2 ( 2 x0 y2 + 3x5 y0 )

= 6 x3 y2 ( 2 y2 + 3x5 )

2. Difference of Squares

Formula:

A2 - B2 = (A + B) (A - B)

Note:

There is no formula for A2 + B2.

e.g.

Factor 81 m8 - 16 y6 z4

81 m8 - 16 y6 z4

= (9 m4)2 - (4 y3 z2)2

Put into the form A2 - B2.

48 981 mmA 2346 416 zyzyB

81 m8 - 16 y6 z4

= (9 m4)2 - (4 y3 z2)2

= (9 m4 + 4 y3 z2) (9 m4 - 4 y3 z2)

A2 + B2 = (A + B) (A - B)

where A = 9 m4 and B = 4 y6 x2

3. Sum / Difference of Cubes

Formulas:

A3 - B3 = (A - B) (A2 + 2AB + B2)

A3 + B3 = (A + B) (A2 - 2AB + B2)

e.g. 1

Factor x3 - 64y3

x3 - 64y3

= (x)3 - (4 y)3

Put into the form A3 - B3

xxA 3 3

yyB 4643 3

x3 - 64y3

= (x)3 - (4 y)3

= (x - 4y) [ x2 + (x) (4y) + (4y)2 ]

A3 - B3 = (A - B) (A2 + AB + B2)

where A = x and B = 4y

x3 - 64y3

= (x)3 - (4 y)3

= (x - 4y) [ x2 + (x) (4y) + (4y)2 ]

= (x - 4y) (x2 + 4xy + 16y2)

e.g. 2

Factor 8x3 + 27y6

8x3 + 27y6

= (2x)3 + (3 y2)3

Put into the form A3 + B3

xxA 283 3

23 6 327 yyB

8x3 + 27y6

= (2x)3 + (3 y2)3

= (2x + 3y2) [ (2x)2 (2x) (3y2) + (3y2)2 ]

A3 + B3 = (A + B) (A2 - AB + B2)

where A = 2x and B = 3y2

8x3 + 27y6

= (2x)3 + (3 y2)3

= (2x + 3y2) [ (2x)2 (2x) (3y2) + (3y2)2 ]

= (2x + 3y2) (4x2 6xy2 + 9y4)

4. Grouping

When there are 4 terms, try grouping:

Group pairs of terms (you may need to rearrange)

Factor each pair

Factor out the common polynomial

e.g.

Factor ac bd + bc ad

ac bd + bc ad

No common factors for each pair.

Thus, we need to rearrange.

ac bd + bc ad

= ac ad + bc bd

ac bd + bc ad

= ac ad + bc bd

= a (c d) + b (c d)

They must have a common factor.

ac bd + bc ad

= ac ad + bc bd

= a (c d) + b (c d)

= (a + b) (c d)

5. Factoring Trinomials

Trinomials are polynomials with 3 terms.

They have the form

Ax2 + Bx + C = 0

We will deal with two cases:

Case 1: A = 1 (By inspection)

Case 2: A ≠ 1 (Decomposition)

Case 1: A = 1 (By inspection)

To factor x2 + Bx + C,

Find 2 numbers that add to B and multiply to C

Simply substitute the numbers into the two

binomial factors

e.g.

Factor x2 + 2x - 15

x2 + 2x - 15

Find two

numbers that ... add to 2

x2 + 2x - 15

Find two

numbers that ... add to 2 and multiply to -15

x2 + 2x - 15

2 numbers:

Sum = 2

Product = -155, -3

x2 + 2x - 15

2 numbers:

Sum = 2

Product = -15

= (x + 5) (x - 3)

Simply sub the numbers in

5, -3

Case 2: A ≠ 1 (Decomposition)

To factor Ax2 + Bx + C,

Find 2 numbers that add to B and multiply to AC

Replace B with these two numbers

Factor by grouping

e.g.

Factor 3x2 - 17x + 10

3x2 - 17x + 10

Find 2 numbers:

Sum = -17

3x2 - 17x + 10

Find 2 numbers:

Sum = -17

Product = 30

3x2 - 17x + 10

Find 2 numbers:

Sum = -17

Product = 30-15, -2

3x2 - 17x + 10

= 3x2 - 15x - 2x + 10Replace B with the two numbers, -2 and -15

3x2 - 17x + 10

= 3x2 - 15x - 2x + 10

= 3x (x - 5) - 2 (x - 5) Factor by grouping

3x2 - 17x + 10

= 3x2 - 15x - 2x + 10

= 3x (x - 5) - 2 (x - 5)

= (x - 5) (3x - 2)

Summary (Factoring methods)

GCF first

Look at the # of terms:

2 terms : - Difference of squares

- Sum / difference of cubes

3 terms: - Inspection (if A = 1)

- Decomposition (if A ≠ 1)

4 terms: - Grouping

Ex. 3

Factor 80 xy3 + 10xz6 completely.

Try this example on your own first.

Then, check out the solution.

80 xy3 + 10xz6

= 10x (8y3 + z6) Factor GCF first.

80 xy3 + 10xz6

= 10x (8y3 + z6)

Don’t stop here.

Do you see what else can be factored?

80 xy3 + 10xz6

= 10x (8y3 + z6)

= 10x [ (2y)3 + (z2)3 ] Sum of cubes

80 xy3 + 10xz6

= 10x (8y3 + z6)

= 10x [ (2y)3 + (z2)3 ]

= 10x (2y + z2) [ (2y)2 - (2y) (z2) + (x2)2 ]

80 xy3 + 10xz6

= 10x (8y3 + z6)

= 10x [ (2y)3 + (z2)3 ]

= 10x (2y + z2) [ (2y)2 - (2y) (z2) + (x2)2 ]

= 10x (2y + z2) (4y2 - 2yz2 + x4)

Ex. 4

Factor x2y - 54 + 6x2 - 9y completely.

Try this example on your own first.Then, check out the solution.

x2y - 54 + 6x2 - 9y

We will factor by grouping (4 terms).

However, we must rearrange so that there will be

common factors.

Can you see how?

x2y - 54 + 6x2 - 9y

= x2y - 9y + 6x2 - 54

This is one way to do so.

x2y - 54 + 6x2 - 9y

= x2y - 9y + 6x2 - 54

= y (x2 - 9) + 6 (x2 - 9)

x2y - 54 + 6x2 - 9y

= x2y - 9y + 6x2 - 54

= y (x2 - 9) + 6 (x2 - 9)

= (x2 - 9) (y + 6)

Don’t stop here.

Can you see what else can be factored?

x2y - 54 + 6x2 - 9y

= x2y - 9y + 6x2 - 54

= y (x2 - 9) + 6 (x2 - 9)

= (x2 - 9) (y + 6)

= (x + 3) (x - 3) (y + 6) Difference of squares

Ex. 5

Factor 3a4 - 7a2 - 20 completely.

Try this example on your own first.Then, check out the solution.

Notice that 3a4 - 7a2 - 20 is a trinomial.

To make it easier to factor, let’s do a substitution.

i.e.

Let x = a2

Then,

3 (a2)2 - 7 (a2) - 20 = 3x2 - 7x - 20

3x2 - 7x - 20

Find 2 numbers:

Sum = -7

Product = -60-12, 5

3x2 - 7x - 20

Find 2 numbers:

Sum = -7

Product = -60

= 3x2 - 12x + 5x - 20

-12, 5

3x2 - 7x - 20

Find 2 numbers:

Sum = -7

Product = -60

= 3x2 - 12x + 5x - 20

= 3x (x - 4) + 5 (x - 4)

-12, 5

3x2 - 7x - 20

Find 2 numbers:

Sum = -7

Product = -60

= 3x2 - 12x + 5x - 20

= 3x (x - 4) + 5 (x - 4)

= (x - 4) (3x + 5)

-12, 5

= (x - 4) (3x + 5)

Finally, we have to back-substitute x = a2:

= (x - 4) (3x + 5)

Finally, we have to back-substitute x = a2:

= (a2 - 4) (3a2 + 5)

Don’t stop here.

Do you see what else can be factored?

= (x - 4) (3x + 5)

Finally, we have to back-substitute x = a2:

= (a2 - 4) (3a2 + 5)

= (a + 2) (a - 2) (3a2 + 5)