Factoring Polynomials

Polynomials:

A polynomial is called an algebraic expression having the power of each variable is a only positive integral numbers.

For example,  8x2 -  16x3y2 + 24 y4 + 10, 9y2 - 16, y2 + 20x + 100.

These examples are a polynomial expression in two variables x and y and polynomial expression in one variable y.

Let us discuss about factoring polynomial .
Factoring Polynomials:

Factoring Polynomials:

Factoring a polynomial is the reverse process of multiplying polynomials.

When we factor a real number, First we are getting for prime factors for real number which multiply together to give the same real number.

For example,   28 = 7 x 2 x 2

When we are going to factor a polynomial,  First we need  to look for simpler polynomial expressions which are multiplied each other and give the same polynomial expression what we started with.

For example,  y5x2 + 25xy = 5xy(x + 5)

Let us solve sample problems on factoring polynomials.
Sample Problems: Factoring Polynomials

Steps to factoring Polynomials:

Step 1: First we factor the polynomial

Step 2: Expand the factorized polynomial using algebraic identities formula.

Step 3: Solve it if possible.

Problem 1:

Factor the polynomial expression: 25m2 – 64

Solution:

The given polynomial expression is 25m2 – 64.

Step 1: Factoring it,

= 52m2 – 82

= (5m)2 - 82

Step 2: Expand it

This is in the form of a2 – b2 = (a + b) (a – b)

So,

= (5x + 8)(5x – 8)

Therefore, the factors are (5x + 8)(5x – 8).

Step 3: Solving it,

5x + 8 = 0

5x = - 8

x = - `8/5`

5x - 8 = 0

5x = 8

x = `8/5`

Therefore, Solutions are x = `8/5` , `- 8/5` .

Problem 2:

Factor the polynomial expression: 9p2 – 81

Solution:

The given polynomial expression is 9p2 – 81.

Step 1: Factoring it,

= 32p2 – 92

= (3p)2 - 92

Step 2: Expand it

This is in the form of a2 – b2 = (a + b) (a – b)

So,

= (3p + 9)(3p – 9)

Therefore, the factors are (3p + 9)(3p – 9).

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Step 3: Solving it,

3p + 9 = 0

3p = - 9

p = -` 9/3` = - 3

3p - 9 = 0

3p = 9

p = `9/3` = 3

Therefore, Solutions are p = 3 , - 3.
Problem 3:

Factor the polynomial expression: A2 – `1/49`

Solution:

The given polynomial expression is A2 – `1/49` .

Step 1: Factoring it,

= A2 – `1/7^2`

= A2 - `(1/7)^2`

Step 2 : Expand it,

This is in the form of a2 – b2 = (a + b) (a – b)

So,

= (A + `1/7` )(A – `1/7` )

Therefore, the factors are (A + `1/7` )(A – `1/7` ).

Step 3 : Solving it,

(A + `1/7` )(A - `1/7` ) = 0

A + `1/7` = 0

A = - `1/7`

X - `1/7` = 0

X = `1/7`

Therefore, Solutions are A = `1/7` , - `1/7` .

Problem 4:

Factor the polynomial expression: y2 + 2y - 15

Solution:

The given polynomial expression is y2 + 2y  - 15.

Step 1 : Factoring it,

y2 + 2y - 15

Step 2 : Expand the polynomial expression

= y2 - 3y + 5y - 15

= y(y - 3) + 5(y - 3)

Taking out common factor,

= (y - 3)(y + 5)

Therefore, the factors are (y + 5)(y – 3).

Step 3 : Solving it,

(y + 5)(y - 3) = 0

y + 5 = 0

y = - 5

y - 3 = 0

y = 3

Therefore, Solutions are y = 3 , - 5.

Problem 5:

Solve for x in the perfect square trinomial: 9x2 + 24x + 16 = 0

Solution:

The given perfect square trinomial is 9x2 + 24x + 16 =0

Step 1: First factoring this perfect square trinomial

(3X)2 + 2·3.4x + 42 = 0

(3x + 4)2 = 0

Step 2: Expand it

This is in the form (ax + b)2 = (ax + b) (ax + b)

(3x + 4)(3x + 4) = 0

Step 3: Solving x

3X + 4 = 0

3X = - 4

Divide by 3 each side.

X = `- 4/3` .