Project in mathematics Submitted by:

Aljan Dada

Submitted to:

Mr.Jenifer Balisacan

LESSON 1:

PRODUCT OF A MONOMIAL BY A POLYNOMIAL

RULE:

IN MULTIPLYING A MONOMIAL BY A POLYNOMIAL THE MULTIPLICATION PROCESS CAN BE ILLUSTRATED

A(X+Y+Z)

=AX+AY+AZ

EXAMPLES 1:

Find (xy + y2)(2x + 9y).

Distribute (xy + y2) over (2x + 9y) to find

(xy + y2)(2x) + (xy + y2)(9y)

Distribute further to get

(xy)(2x) + (y2)(2x) + (xy)(9y) + (y2)(9y)

and then simplify for

2x2y + 2xy2 + 9xy2 + 9y3

which can be combined further to give us

2x2y + 11xy2 + 9y3.

EXAMPLE 2:Find the product (4x2 + 10x)(2x + 3).

Let's FOIL it up.

The product of the first terms is

(4x2)(2x)=8x3

The product of the outer terms is

(4x2)(3)=12x2

The product of the inner terms is

(10x)(2x)=20x2

The product of the last terms is

(10x)(3)=30x

Adding the little products

8x3 + 12x2 + 20x2 + 30x

which simplifies to

8x3 + 32x2 + 30x.

Example 3:Find the product (2xy - y)(3x2 + 7xy).

Using FOIL, we find

(2xy)(3x2) + (2xy)(7xy) + (-y)(3x2) + (-y)(7xy)

This simplifies to

6x3y + 14x2y2 - 3x2y - 7xy2

Lesson 2:

Product of two binomials

Say we want to find the product ( x – 3)(2x + 1).

RULE: Multiply the First terms in each bracket: x times 2x = 2x². Multiply Outer terms: (x times +1) and Inner terms (–3 times 2x), then add = – 5x. Multiply the Last terms in each bracket: – 3 times +1 = – 3.

So, ( x – 3) (2x + 1) = 2x² – 5x – 3.

EXAMPLE1:

Example: can you work out which binomials to multiply to get 4x2 − 9

4x2 is (2x)2, and 9 is (3)2, so we have:

4x2 − 9 = (2x)2 − (3)2

And that can be produced by the difference of squares formula:

(a+b)(a−b) = a2 − b2

Like this ("a" is 2x, and "b" is 3):

(2x+3)(2x−3) = (2x)2 v (3)2 = 4x2 − 9

So the answer is that we can multiply (2x+3) and (2x−3) to get 4x2 − 9

LESSON 3:

SQUARE OF A BINOMIAL

Example 1. 

  Square the binomial (x + 6).

Solution.    (x + 6)2 = x2 + 12x + 36x2 is the square of x.12x  is  twice the product of x with 6.

(x· 6 = 6x.  Twice that is 12x.)36 is the square of 6.

x2 + 12x + 36 is called a perfect square trinomial.  It is the square of a binomial.

EXAMPLE2:

Is this a perfect square trinomial:  x2 + 14x + 49 ?

Answer.   Yes.  It is the square of (x + 7).x2 is the square of x.  49 is the square of

7.  And 14x is twice the product of x· 7.In other words, x2 + 14x + 49 could

be factored asx2 + 14x + 49 = (x + 7)2

Note:  If the coefficient of x had been any number but 14, this would not have been a perfect square trinomial.

LESSON 4:

PRODUCT OF SUM AND DIFFERENCE OF TWO THE SAME TERMS

RULE:

TO FIND THE PRODUCT OF THE SUM AND DIFFERENCEOF THE TWO TERMS.

1.MULTIPLYTHE FISST TERM;PLUS

2.MULTIPLY THE LAST TERM.

Example 1: (x – 4)(x + 4)

You can use the shortcut to do these special distributions.

The second term will always be negative, and a perfect square like the first term: (–4)(+4) = –16.

Example 2: (ab – 5)(ab + 5)

Try the same easy process — multiplying the sum of two terms with their difference — with this slightly more complicated, variable term.

The second term is negative, and a perfect square like the first term: 5 = –25.

LESSON 5:

SQUARE OF A

TRINOMIAL

RULE:

1.SQUARE THE FIRST TERM;PLUS

2.SQUARE THE SECOND TERM;PLUS

3.SQUARE THE LAST TERM;PLUS

4.TWICE THE PRODUCT OF THE FIRSTAND LAST TERM

5.TWICE THE FIRST AND LAST TERM;PLUS

6.TWICE THE SECOND AND LAST TERM

EXAMPLES

. Example 1.  

 Square the binomial (x + 6).

Solution.    (x + 6)2 = x2 + 12x + 36x2 is the square of x.12x  is  twice the product of x with 6.  (x· 6 = 6x.  Twice that is 12x.)

36 is the square of 6.

x2 + 12x + 36 is called a perfect square trinomial.  It is the square of a binomial.

Example 2.  

 Is this a perfect square trinomial:  x2 + 14x + 49 ?

Answer.   Yes.  It is the square of (x + 7).x2 is the square of x.  49 is the square of

7.  And 14x is twice the product of x· 7.In other words, x2 + 14x + 49 could

be factored asx2 + 14x + 49 = (x + 7)2

Note:  If the coefficient of x had been any number but 14, this would not have been a perfect square trinomial.

Example 2.   Square the binomial (3x − 4).

Solution.    (3x − 4)2 = 9x2 − 24x + 169x2 is the square of 3x.−24x  is  twice the product of  3x· −4.

(3x· −4 = −12x.  Twice that is −24x.)

16 is the square of −4.

LESSON 6:

PRODUCT OF SPECIAL

CASEOF A BINOMIAL

AND A TRINOMIAL

The square of a binomial is always the sum of:

1. The first term squared,2. 2 times the product of the first and second terms, and3. the second term squared.

When a binomial is squared, the resulting trinomial is called a perfect square trinomial.

Examples:

1.(x + 5)2 

=x 2 + 10x + 25 

2.(100 - 1)2 

2 = 10000 - 200 + 1

3.(2x - 3y)2

 = (2x)2 +2(2x)(- 3y) + (- 3y)2

 = 4x 2 -12xy + 9y 2 

4.(2x - y)(2x + y)

= (2x)2 - y 2

 = 4x 2 - y 2 

LESSON 7:

CUBEOF A BINOMIAL

RULE:

1.CUBE THE FIRST TERM OF A BINOMIAL;PLUS

2.TRICE THE PRODUCT OF THE SQUARE OF THE FIRST TERM AND THE LAST TERM

3.TRICE THE PRODUCT OF THE FIRST TERM AND SQUARE OF THE FIRST TERM.

4.CUBE THE LAST LAST OF THE BINOMIAL

Solution:(a+ b)3 =(a)3+3(a)2 (b)+3(a) (b)2+b3

EXAMPLES

1.(3x+4)3

=(3x)3+3(3x)2 (4)+3(3x) (4)2+43

= 27x3+108x2+144x+64

2.(4x+ 6)3

=(4x)3+3(4x)2 (6)+3(4x) (6)2+63

= 64x3+288x2+432x+216

3.(3x-4)3

= (3x)3-3(3x)2 (4)+3(3x) (4)2-43

= 27x3-108x2+144x-64

.

LESSON 8:

APPLICATIONS OF SPECIAL PRODUCT

EXAMPLE:

ONE CAT CARRIES HETEROZYGONS LONG HAIRED TRAITS (SS)AND ITS MATE CARRIES HETEROZYGONS LONG HAIRED TRAITS (Ss).

1.WHAT IS THE CHANCES THAT THE OFFSPRINGS IS A LONG HAIRED CAT?SHORT HAIRED CAT?

SOLVE.

LONG HAIRED CAT -75 OR75%

SHORTHAIRED CAT 25 OR 25%

LESSON 9:

FACTORING POLYNIMIALS WITH

COMMON MONOMIAL FACTOR

FACTORING= REVERSE PROCESS OF MULTIPLICATION

=PROCESS OF GETTING FACTORS OF A GIVEN EXPRESSION(POLYNOMIAL).

EXAMPLE:

1.2,4,8

SOL.

2 2,4,8

1,2,4

2.54,18

9 54,18

2 6,-2

3,-1

LESSON 10:

FACTORING THE DIFFERENCE

OF TWO SQUARES

This expression is called a difference of two squares.(Notice the subtraction sign between the terms.)

 You may remember seeing expressions like this one when you worked with multiplying algebraic expressions.  Do you remember ...

 

If you remember this fact, then you already know that:

 The factors of  

are and

Example 1:

Factor:   x2 - 9

Both x2 and 9 are perfect squares.  Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give x2 ?  The answer is x.What times itself will give 9 ?  The answer is 3.

These answers could also be negative values, but

positive values will make our work easier.

The factors are (x + 3) and (x - 3).Answer:  (x + 3) (x - 3)  or   (x - 3) (x + 3)   (order is not important)