2.5 Special Products

In the previous sections, the distributive property was used in multiplying polynomials. For example, the product of two binomials and using the distributive property is as follows:

Take note that a shortcut for finding the product of two binomials is to add the products of four pairs of terms of the binomials namely: the product of the first terms, the product of the outer terms, the product of the inner terms, and the product of the last terms. This special order or method of multiplying binomials is known as the FOIL method. The word FOIL is formed from the first letters of the products of the terms of the two binomials to be added.

The following table illustrates the FOIL method of finding the product of two binomials, 2x + 1 and 3x – 5.

FOIL MethodF stands for the product of the First terms

(2x + 1) (3x – 5) ( 2x) (3x) = 6x2 F

O stands for the product of the Outer terms

(2x + 1) (3x – 5 ) ( 2x) (–5 ) = –10x O

I stands for the product of the Inner terms

(2x + 1) (3x – 5 ) (1) (3x ) = 3x I

L stands for the product of the Last terms

(2x + 1) (3x – 5 ) (1) (–5) = –5 L

F O I L

The product of Combine similar terms

When using the FOIL method in finding certain types of products, a specific pattern is observed. Such patterns lead to the special product formulas and can be used to find the products of binomials which are called special products. The following are the different types of special products.

Types of Special Products

A. Product of the Sum and Difference of Two Numbers

The product of the sum and difference of two numbers is the difference of two squares. The difference of two squares is the square of the 1st number minus the square of the 2nd number. Thus,

Example. Find the product of the following binomials.

1

=

=

=

B. Square of a Binomial

The square of a binomial is a perfect trinomial square. A perfect trinomial square is the square of the first number plus or minus twice the product of the two numbers plus the square of the second number.

Example. Find the product of the following binomial.

C. Product of Any Two Binomials

The product of any two binomials is a general trinomial and is obtained by using the FOIL method.

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Example. Find the product of the following binomials.

D. Cube of a Binomial

The cube of a binomial is a quadrinomial with the following terms:1st term is the cube of the 1st number 2nd term is plus or minus thrice the square of the first number times the second number3rd term is plus thrice the first number times the square of the second number4th term is plus or minus the cube of the second number

Example. Find the product of each of the following binomials.

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E. Product of Binomial and Trinomial

The product of a binomial and a trinomial in the form is the sum or difference of two cubes.

Example. Find the product of each of the following. 1)

2)

F. Product of Trinomials and Quadrinomials

The special product formulas can be used to find the products of trinomials and quadrinomials by grouping the terms into binomials.

Example 1. Find the product of the following trinomials. a)

After grouping, the 1st term is 3a and the 2ndsecond terms are ±(2c –3d) for both binomials. Use the formula for the product of the sum and difference of two numbers

Expand . Use the formula for

the square of a binomial.

Simplify

Another solution of Example a:

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a) After grouping, the 1st two terms

and are the 1st terms of the binomials. The 3rd terms, ±3d are the 2nd

terms of binomials. Use the FOIL method Multiply the 1st term

. Use the formula for the product of the sum and difference of two numbers. Combine similar terms

Note: In Example a, both the formula for the product of the sum and difference of two numbers and the FOIL method can be used.

b) After grouping, (5m + 2n) and (5m – 2n) are the 1st terms while –5p and –3p are

the 2nd terms of the binomials. Use the FOIL method.

Multiply the 1st term (5m+2n)(5m-2n).

Use the formula for the product of the sum and difference of two numbers. Combine similar terms

Note: Unlike Example a, Example b can be expressed only as the product of binomials after grouping. Thus, the FOIL method was used.

c)

After grouping, the 1st term is 6x and

the 2nd term is –(4y + 5z). Use the formula for the square of a binomial ( square of the difference of

two numbers ). Expand (4y+5z)2. Use the formula for the square of the sum of two numbers. Simplify

Note: The square of a trinomials can always be expressed as the square of a binomial after grouping the terms of the given trinomial. Thus, the formula for the square of a binomial is used.

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Example 2. Find the product of the following quadrinomials.

a) After grouping, the 1st term is (2x + 3y) and the 2nd term is ±(z – 4) for both binomials. Use

the formula for the product of the sum and difference of two numbers.

Expand (2x + 3y)2 and (z - 4)2. Use the formula for the square of a binomial.

Simplify

Note: If after grouping, the product of the given quadrinomials cannot be expressed as the product of the sum and difference of two expressions , the FOIL method is used.

b)

After grouping, the 1st term is (4a–3y and the 2nd term is –(2z – 5). Use the formula for the square of a binomial. Expand (4a – 3y)2 and (2z – 5)2. Use the formula for the square of a binomial. Multiply (4a – 3y) and (2z – 5). Use FOIL method.

Simplify

Note: The square of a quadrinomial can always be expressed as the square of a binomial after grouping the terms of the given quadrinomial. Thus the formula for the square of a binomial is used.

G. Square of a Polynomial

The square of a polynomial is equal to the sum of the squares of each term of the polynomial and twice the product of any combination of two terms. This method of finding the square of a polynomial is useful if the polynomial contains more than four terms.

Example: Find the product of the following polynomials.

1)

6

2)

Summary of the Special Product Formulas.

Special Product Formulas

Product of the Sum and Difference of Two Numbers = Difference of Two Squares

Square of a Binomial = Perfect Trinomial Square

Product of any Two Binomials = General Trinomial

Cube of a Binomial = Quadrinomial

Product of Binomial and Trinomial

Square of a Polynomial

The square of a polynomial is equal to the sum of the squares of each term of the polynomial and twice the product of any combination of two terms.

2.6 Factoring

Factoring is the reverse of multiplication. It is the process of expressing a given polynomial as a product of its factors.

A polynomial with integral coefficients is said to be prime if it has no monomial or polynomial factors with integral coefficients other than itself and one. Thus, a polynomial with integral coefficients is said to be completely factored when each of its polynomial factors is prime.

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Factor out HCF, 5p2q3r

Types of Factoring

A. Removal of the Highest Common Factor (HCF) A common factor is a factor contained in every term of a polynomial.

The highest common factor is the product of the greatest common factors of the numerical coefficients and the literal coefficients having the least exponents in every term of a polynomial.

Example: The HCF of a) 5x and 15x2 is 5x b) 3ab2 and 6a is 3a c) 8x2y2z3, 16x3y3z2 and 24x4yz4 is 8x2yz2

If every term of a polynomial contains a common factor, then the polynomial can be factored by removing the HCF. Thus, the factors are the HCF and the quotient obtained by dividing the given polynomial by the HCF.

Example. Factor the following completely:

B. Difference of Two Squares

The factors of the difference of two squares are the sum and difference of the square roots of the two squares.

Note: The sum of two squares (a2 + b2) is a prime polynomial, hence, it is not factorable. Example. Factor the following completely.

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Multiply 3y(h – a) by -1

Factor out HCF, 8xy

Factor out HCF, 3(a – h)

Simplify

Factor out (5a – 2)

Simplify

Simplify

C. Sum and Difference of Two Cubes

The factors of the sum or difference of two cubes consist of a binomial factor and a trinomial factor.

The binomial factor is the sum or difference of the cube roots of the two terms of the given binomial.

The trinomial factor is obtained from the binomial factor and has the following terms:

First term is the square of the 1st term of the binomial factor. Middle term is the product of the two terms of the binomial factor multiplied by (–1). Last term is the square of the 2nd term of the binomial factor.

Example. Factor the following completely.

Factor out HCF, 9a2b2

Factor out HCF, 8

9

Factor out HCF, b

Factor out HCF, 3

Factor out HCF, 3y2x

Factor out HCF, 2a

Factor (4a2 – b2) as the difference of two squares

Factor (x4 - y4) as thedifference of two squares

Factor as the difference of two squares

Factor as the sum and difference of two cubes

Factor as the differenceof two cubes

Factor (x2 – y2) as thedifference of two squares

Factor (x2 – y2) as the difference of two squares

Note: Example 7 can also be factored as the difference of two squares

D. Perfect Trinomial Square (PTS)

A trinomial is a perfect trinomial square if the first and last terms are perfect squares and the middle term is plus or minus twice the product of the square roots of the 1 st and the last terms.

The factors of a PTS are the square of the sum or the difference of the square root of the 1st and the last terms of the given trinomial.

Example. Factor the following completely.

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Factor out HCF, 2b

Factor out HCF, 3a

Factor as difference of two squares

Simplify

Simplify

Factor as the sum of two cubes

Factor out HCF, 9x2

Combine similar terms

Factor out HCF, –2

Simplify

Factor out HCF, 4y

Apply distributive property

E. General Trinomial

A general trinomial of the form can be factored into the product of two binomials . a, b, c, d can be obtained by using the trial and error method.

Trial: a) The first terms of both binomials are factors of the first term of the given trinomial.

b) The second terms of both binomials are factors of the last term of the given trinomial.

To check if trial is correct, the sum of the products of the outer and inner terms must be equal to the middle term of the given trinomial.

Example. Factor the following completely.

11

Factor out HCF, 3

Simplify

Factor out HCF, (3 + x)

Factor as the differenceof two square

Simplify

Factor the 1st first grouped terms as a PTS

Factor as the difference of two squares terms as a PTSSimplify

Factor out HCF of each grouped terms

Simplify

Factor each grouped terms as a PTS

Simplify

F. Factoring by Grouping

Factoring by grouping is used to factor polynomials consisting of more than three terms. In factoring by grouping, the terms of the given polynomial are grouped to form a

binomial or a trinomial that are both factorable.

Example. Factor the following completely.

12

Factor the 2nd grouped terms as a PTS

Factor as the difference of two squares terms as a PTSSimplifyFactor the 1st grouped terms as a PTS

Factor out HCF, (x -2y)

Factor out HCF, (5a – 6b)

Factor the 1st grouped terms as difference of two squares and the 2nd grouped terms by removal of HCF, 2

Factor the 1st groupedterms as difference of two cubes and the 2nd grouped terms by removal of HCF, 3x

Simplify

Factor out HCF, (2x –y)

Factor the 1st Grouped terms as a PTS and the 2nd grouped terms by removal of HCF, 6y

Factor as a PTS

Simplify

G. Factoring by Completing the Square

Factoring by completing the square is applicable to binomials whose terms are both perfect squares and to trinomials with at least two terms that are perfect squares. The method consists of adding and subtracting a term that is a perfect square that will make the given binomial or the trinomial a PTS. The resulting expression is then factored as the difference of two squares.

Example. Factor the following completely.

13

Factor TheoremA polynomial P(x) has a factor x – c, where c is any real number if and only if P(c) = 0 (remainder is 0).

H. Factoring by Synthetic Division

Factoring by synthetic division is applicable to a polynomial in one variable whose degree is higher than two.

Factor Theorem The factor theorem is used to determine if a binomial x – c is a factor of the given polynomial.

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Example 1. Determine if:

a) is a factor of Replace c with –1 Perform the indicated operations

Since P(-1) = 0, then x + 1 is a factor of x3 + 2x2 – 5x –6.

b) is a factor of Replace c with 3 = 27 + 27 – 15 –15 Perform the indicated operation

and simplify.

Since P(3) = 24, y – 3 is not a factor of y3 + 3y2 – 5y – 15.

Example 2. Factor the following using synthetic division.

1)

The factors of are

2)

The factors of are

3)

15

1

The factors of are

4)

The factors of are

I. Factoring the Sum or Difference of Two Prime Odd Powers

The sum or difference of two prime odd powers can be performed using the following formulas:

Example. Factor the following completely.

GENERAL PROCEDURE FOR FACTORING1. Factor out any common factor.2. If the polynomial is a binomial, factor it as the difference of two squares (a2 – b2) or the sum or

difference of two cubes (a3 b3). The sum of two squares (a2 + b2) is prime.

16

5

4

3

3

2

2

3. If the polynomial is a trinomial, factor it as a PTS (a2 2ab + b2) or by trial and error method.4. If the polynomial has more than three terms, try factoring by grouping.5. If the polynomial is a binomial whose terms are both perfect squares or a trinomial with at least two

terms that are perfect squares but is not a PTS, use factoring by completing the square.

6. If the polynomial is in one variable and has a degree higher than two, use synthetic division 7. If the binomial is the sum or difference of two prime odd powers, apply the formulas 8. Check if all the factors are prime.

Summary of Special types of Factoring

Special Types of Factoring

Removal of the Highest Common Factor (HCF)

Difference of Two Squares Sum and Difference of Two Cubes Squares

Perfect Trinomial Square (PTS)

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