# Polynomial

## Summary

- Ascending Order:Example

x^{0}, x^{1}, x^{2}, x^{3}, ... - Descending Order:Example

..., x^{3}, x^{2}, x^{1}, x^{0} - Adding and Subtracting PolynomialsExample

→ add and subtract like terms. - a(x + y + z)= ax + ay + az
- (a + b)(x + y)= ax + ay + bx + by
- (a + b)(x + y + z)= ax + ay + az+bx + by + bz
- (a + b)
^{2}= a^{2}+ 2ab + b^{2} - (a - b)
^{2}= a^{2}- 2ab + b^{2} - (a + b)(a - b)= a
^{2}- b^{2} - (a + b)
^{3}= a^{3}+ 3a^{2}b + 3ab^{2}+ b^{3} - (a - b)
^{3}= a^{3}- 3a^{2}b + 3ab^{2}- b^{3} - ax + aya= x + y
- Long Division:Example

f(x) ÷ (x - a)

## Ascending Order

### Example

Arrange the terms of the given polynomial so that the powers of x are in ascending order.

x

x

^{2}+ 3y^{2}- 9xy - 4x^{3}#### Solution

- 1x
^{2}+ 3y^{2}- 9xy - 4x^{3} - 2= 3y
^{2}- 9xy + x^{2}- 4x^{3}

See the exponents of x.

First write the x

(x

Write the x

Write the x

And write the remaining term: -4x

First write the x

^{0}term: 3y^{2}.(x

^{0}term: doesn't have x)Write the x

^{1}term: -9xy.Write the x

^{2}term: +x^{2}.And write the remaining term: -4x

^{3}.### Example

Arrange the terms of the given polynomial so that the powers of y are in ascending order.

x

x

^{2}+ 3y^{2}- 9xy - 4x^{3}#### Solution

- 1x
^{2}+ 3y^{2}- 9xy - 4x^{3} - 2= -4x
^{3}+ x^{2}- 9xy + 3y^{2}

See the exponents of y.

First write the y

(y

Write the y

And write the remaining term: +3y

First write the y

^{0}terms: -4x^{3}+ x^{2}.(y

^{0}terms: don't have x)Write the y

^{1}term: -9xy.And write the remaining term: +3y

^{2}.## Descending Order

### Example

Arrange the terms of the given polynomial so that the powers of x are in descending order.

x

x

^{2}+ 3y^{2}- 9xy - 4x^{3}#### Solution

- 1x
^{2}+ 3y^{2}- 9xy - 4x^{3} - 2= -4x
^{3}+ x^{2}- 9xy + 3y^{2}

See the exponents of x.

First write the x

Write the x

Write the x

And write the remaining x

(x

First write the x

^{3}term: -4x^{3}.Write the x

^{2}term: +x^{2}.Write the x

^{1}term: -9xy.And write the remaining x

^{0}term: +3y^{2}.(x

^{0}term: doesn't have x)### Example

Arrange the terms of the given polynomial so that the powers of y are in descending order.

x

x

^{2}+ 3y^{2}- 9xy - 4x^{3}#### Solution

- 1x
^{2}+ 3y^{2}- 9xy - 4x^{3} - 2= 3y
^{2}- 9xy -4x^{3}+ x^{2}

See the exponents of y.

First write the y

Write the y

And write the remaining y

(y

First write the y

^{2}term: 3y^{2}.Write the y

^{1}term: -9xy.And write the remaining y

^{0}terms: -4x^{3}+ x^{2}.(y

^{0}terms: don't have x)## Adding and Subtracting Polynomials

### Example

Simplify the given expression.

7x

7x

^{2}- 3x + 2x^{2}+ 8x#### Solution

- 17x
^{2}- 3x + 2x^{2}+ 8x - 2= (7 + 2)x
^{2}+ (-3 + 8)x - 3= 9x
^{2}+ 5x

1:

Like terms:

terms that have the same variable(s) and exponent(s).

In this polynomial,

7x

because they have the same x

By the same way,

-3x and +8x are like terms,

because they have the same x (= x

terms that have the same variable(s) and exponent(s).

In this polynomial,

7x

^{2}and +2x^{2}are like terms,because they have the same x

^{2}.By the same way,

-3x and +8x are like terms,

because they have the same x (= x

^{1}).1 ~ 2:

Only like terms can be added or subtracted

by adding or subtracting their coefficients.

7x

So 7x

-3x and +8x are like terms.

So -3x + 8x = (-3 + 8)x.

by adding or subtracting their coefficients.

7x

^{2}and +2x^{2}are like terms.So 7x

^{2}+ 2x^{2}= (7 + 2)x^{2}.-3x and +8x are like terms.

So -3x + 8x = (-3 + 8)x.

### Example

Simplify the given expression.

3a

3a

^{2}+ 2ab + ab^{2}- 6ab + 9ab^{2}- a^{2}#### Solution

- 13a
^{2}+ 2ab + ab^{2}- 6ab + 9ab^{2}- a^{2} - 2= (3 - 1)a
^{2}+ (1 + 9)ab^{2}+ (2 - 6)ab - 3= 2a
^{2}+ 10ab^{2}- 4ab

1 ~ 2:

3a

So 3a

+ab

So +ab

+2ab and -6ab are like terms.

So +2ab - 6ab = +(2 - 6)ab.

ab

There's a reason.

Usually, we write the terms in descending order.

Both ab

But ab

So you should write the ab

before the ab term.

^{2}and -a^{2}are like terms.So 3a

^{2}- a^{2}= (3 - 1)a^{2}.+ab

^{2}and +9ab^{2}are like terms.So +ab

^{2}+ 9ab^{2}= +(1 + 9)ab^{2}.+2ab and -6ab are like terms.

So +2ab - 6ab = +(2 - 6)ab.

ab

^{2}term is written before ab term.There's a reason.

Usually, we write the terms in descending order.

Both ab

^{2}and ab have the same power a.But ab

^{2}has the higher power b: b^{2}.So you should write the ab

^{2}termbefore the ab term.

## Multiplying a Monomial and a Polynomial

### Example

Simplify the given expression.

x(3x

x(3x

^{2}- 5x + 8)#### Solution

- 1x(3x
^{2}- 5x + 8) - 2= x⋅3x
^{2}+ x⋅(-5x) + x⋅8 - 3= 3x
^{3}- 5x^{2}+ 8x

1 ~ 2:

Multiply x and each term of (3x

^{2}- 5x + 8).2 ~ 3:

### Example

Simplify the given expression.

(5 - a + 3b)a - 3(a

(5 - a + 3b)a - 3(a

^{2}+ ab - 2)#### Solution

- 1(5 - a + 3b)a - 3(a
^{2}+ ab - 2) - 2= 5⋅a - a⋅a + 3b⋅a - 3⋅a
^{2}- 3⋅ab - 3⋅(-2) - 3= 5a - a
^{2}+ 3ab - 3a^{2}- 3ab + 6 - 4= -4a
^{2}+ 5a + 6

1 ~ 2:

Multiply each term of (5 - a + 3b) and a.

And multiply -3 and each term of (a

And multiply -3 and each term of (a

^{2}+ ab - 2).3 ~ 4:

## FOIL Method

### Example

Simplify the given expression.

(x - 2)(x - 3)

(x - 2)(x - 3)

#### Solution

- 1(x - 2)(x - 3)
- 2= x⋅x + x⋅(-3) - 2⋅x - 2⋅(-3)
- 3= x
^{2}- 3x - 2x + 6 - 4= x
^{2}- 5x + 6

1 ~ 2:

The FOIL method is a way

to multiply two binomials.

First multiply the First two terms.

x⋅x = x

Multiply the Outer terms.

+x⋅(-3) = -3x

Multiply the Inner terms.

-2⋅x = -2x

And multiply the Last two terms.

-2⋅(-3) = +6

to multiply two binomials.

First multiply the First two terms.

x⋅x = x

^{2}Multiply the Outer terms.

+x⋅(-3) = -3x

Multiply the Inner terms.

-2⋅x = -2x

And multiply the Last two terms.

-2⋅(-3) = +6

3 ~ 4:

-3x - 2x = -5x

## Multiplying Polynomials

### Example

Simplify the given expression.

(x - 3)(x

(x - 3)(x

^{2}- x + 7)#### Solution

- 1(x - 3)(x
^{2}- x + 7) - 2= x⋅x
^{2}+ x⋅(-x) + x⋅7 - 3-3⋅x
^{2}- 3⋅(-x) - 3⋅7 - 4= x
^{3}- x^{2}+ 7x - 5-3x
^{2}+ 3x - 21 - 6= x
^{3}- 4x^{2}+ 10x - 21

1, 2:

Multiply x and each term of (x

^{2}- x + 7).1, 3:

And multiply -3 and each term of (x

^{2}- x + 7).4 ~ 6:

### Example

Simplify the given expression.

(x

(x

^{2}+ 7x - 1)(3x^{2}- x + 8)#### Solution

- 1(x
^{2}+ 7x - 1) - 2= x
^{2}⋅x^{2}+ x^{2}⋅(-x) + x^{2}⋅8 - 3+7x⋅x
^{2}+ 7x⋅(-x) + 7x⋅8 - 4-x⋅x
^{2}- x⋅(-x) - x⋅8 - 5= 3x
^{4}- x^{3}+ 8x^{2} - 6+21x
^{3}- 7x^{2}+ 56x - 7-3x
^{2}+ x - 8 - 8= 3x
^{4}+ 20x^{3}- 2x^{2}+ 57x - 8

1, 2:

Multiply x

^{2}and each term of (3x^{2}- x + 8).1, 3:

Multiply +7x and each term of (3x

^{2}- x + 8).1, 4:

And multiply -1 and each term of (3x

^{2}- x + 8).5 ~ 8:

-x

+8x

= (+8 - 7 - 3)x

= -2x

+56x + x = +57x

^{3}+ 21x^{3}= +20x^{3}+8x

^{2}- 7x^{2}- 3x^{2}= (+8 - 7 - 3)x

^{2}= -2x

^{2}+56x + x = +57x

## Square of a Sum: (a + b)^{2}

### Formula

(a + b)

^{2}= a

^{2}+ 2ab + b^{2}The middle sign in (a + b)

So the middle term, (+)2ab, is also (+).

^{2}is (+).So the middle term, (+)2ab, is also (+).

### Example

Simplify the given expression.

(x + 3)

(x + 3)

^{2}#### Solution

- 1(x + 3)
^{2} - 2= x
^{2}+ 2⋅x⋅3 + 3^{2} - 3= x
^{2}+ 6x + 9

1 ~ 2:

The middle sign in (x + 3)

So the middle term, +2⋅x⋅3,

is also (+).

^{2}is (+).So the middle term, +2⋅x⋅3,

is also (+).

## Square of a Difference: (a - b)^{2}

### Formula

(a - b)

^{2}= a

^{2}- 2ab + b^{2}The middle sign in (a - b)

So the middle term, (-)2ab, is also (-).

^{2}is (+).So the middle term, (-)2ab, is also (-).

### Example

Simplify the given expression.

(x - 7)

(x - 7)

^{2}#### Solution

- 1(x - 7)
^{2} - 2= x
^{2}- 2⋅x⋅7 + 7^{2} - 3= x
^{2}- 14x + 49

1 ~ 2:

The middle sign in (x - 7)

So the middle term, -2⋅x⋅7,

is also (-).

^{2}is (-).So the middle term, -2⋅x⋅7,

is also (-).

## Product of a Sum and a Difference: (a + b)(a - b)

### Formula

(a + b)(a - b)

= a

^{2}- b^{2}### Example

Simplify the given expression.

(x + 2)(x - 2)

(x + 2)(x - 2)

#### Solution

- 1(x + 2)(x - 2)
- 2= x
^{2}- 2^{2} - 3= x
^{2}- 4

### Example

Find the value of the given expression.

(Use the above formula.)

103⋅97

(Use the above formula.)

103⋅97

#### Solution

- 1103⋅97
- 2= (100 + 3)(100 - 3)
- 3= 100
^{2}- 3^{2} - 4= 10000 - 9
- 5= 9991

1 ~ 2:

103 = 100 + 3

97 = 100 - 3

97 = 100 - 3

2 ~ 3:

Use the formula.

### Example

Simplify the given expression.

(x

(x

^{2}+ 1)(x + 1)(x - 1)#### Solution

- 1(x
^{2}+ 1)(x + 1)(x - 1) - 2= (x
^{2}+ 1)(x^{2}- 1^{2}) - 3= (x
^{2}+ 1)(x^{2}- 1) - 4= (x
^{2})^{2}- 1^{2} - 5= x
^{4}- 1

1 ~ 2:

Use the formula.

2 ~ 3:

-1

^{2}= -13 ~ 4:

Use the formula again.

4 ~ 5:

## Cube of a Sum: (a + b)^{3}

### Formula

(a + b)

^{3}= a

^{3}+ 3a^{2}b + 3ab^{2}+ b^{3}The middle sign in (a + b)

So the signs of the right side terms

are all (+).

^{3}is (+).So the signs of the right side terms

are all (+).

### Example

Simplify the given expression.

(x + 2)

(x + 2)

^{3}#### Solution

- 1(x + 2)
^{3} - 2= x
^{3}+ 3⋅x^{2}⋅2 + 3⋅x⋅2^{2}+ 2^{3} - 3= x
^{3}+ 6x^{2}+ 12x + 8

1 ~ 2:

The middle sign in (x + 2)

So the middle term, +2⋅x⋅3,

is also (+).

^{3}is (+).So the middle term, +2⋅x⋅3,

is also (+).

## Cube of a Difference: (a + b)^{3}

### Formula

(a - b)

^{3}= a

^{3}- 3a^{2}b + 3ab^{2}- b^{3}The middle sign in (a - b)

So the signs of the right side terms

alternate:

(+), (-), (+), (-).

^{3}is (-).So the signs of the right side terms

alternate:

(+), (-), (+), (-).

### Example

Simplify the given expression.

(x - 5)

(x - 5)

^{3}#### Solution

- 1(x - 5)
^{3} - 2= x
^{3}- 3⋅x^{2}⋅5 + 3⋅x⋅5^{2}- 5^{3} - 3= x
^{3}- 15x^{2}+ 75x + 8

1 ~ 2:

The middle sign in (x - 5)

So the signs of the right side terms

alternate:

(+), (-), (+), (-).

^{3}is (-).So the signs of the right side terms

alternate:

(+), (-), (+), (-).

## Dividing a Polynomial by a Monomial

### Example

Simplify the given expression.

5x

^{2}- 3xx

#### Solution

- 15x
^{2}- 3xx - 2=5x
^{2}x-3xx - 3= 5x - 3

1 ~ 2:

Divide each term by the denominator x.

2 ~ 3:

### Example

Simplify the given expression.

12x

^{2}y + 4xy^{3}2xy

#### Solution

- 112x
^{2}y + 4xy^{3}2xy - 2=12x
^{2}y2xy+4xy^{3}2xy - 3= 6x + 2y
^{2}

1 ~ 2:

Divide each term by the denominator 2xy.

2 ~ 3:

12x

= [12/2]⋅x

= 6x

+4xy

= [+4/2]⋅[x/x]⋅y

= +2y

^{2}y/2xy= [12/2]⋅x

^{2 - 1}⋅[y/y]= 6x

+4xy

^{3}/2xy= [+4/2]⋅[x/x]⋅y

^{3 - 1}= +2y

^{2}### Example

Simplify the given expression.

6x

^{5}+ 3x^{4}- 12x^{2}3x

^{2}#### Solution

- 16x
^{5}+ 3x^{4}- 12x^{2}3x^{2} - 2=6x
^{5}3x^{2}+3x^{4}3x^{2}-12x^{2}3x^{2} - 3= 2x
^{3}+ x^{2}- 4

1 ~ 2:

Divide each term by the denominator 3x

^{2}.2 ~ 3:

6x

= [6/3]⋅x

= 2x

+3x

= [+3/3]⋅x

= +x

-12x

= [-12/3]⋅[x

= -4

^{5}/3x^{2}= [6/3]⋅x

^{5 - 2}= 2x

^{3}+3x

^{4}/3x^{2}= [+3/3]⋅x

^{4 - 2}= +x

^{2}-12x

^{2}/3x^{2}= [-12/3]⋅[x

^{2}/x^{2}]= -4

## Long Division

### Example

Simplify the given expression.

(Use the long division.)

(Use the long division.)

x

^{2}+ 3x - 10x - 2

#### Solution

- 1x+5
- 2x - 2x
^{2}+ 3x - 10 - 3x
^{2}- 2x - 45x - 10
- 55x - 10
- 60
- 7(given) = x + 5

2:

Just like dividing numbers,

draw the division form like this.

Write the numerator (x

in the form.

And write the denominator x - 2

in the left side of the form.

draw the division form like this.

Write the numerator (x

^{2}+ 3x - 10)in the form.

And write the denominator x - 2

in the left side of the form.

1, 2, 3:

The goal is to remove x

by using x - 2.

You can make x

by multiplying x - 2 and x.

So write x

on the top of x

And multiply x - 2 and x.

(x - 2)⋅x = x

Write this under x

^{2}of (x^{2}+ 3x - 10)by using x - 2.

You can make x

^{2}by multiplying x - 2 and x.

So write x

on the top of x

^{2}.And multiply x - 2 and x.

(x - 2)⋅x = x

^{2}- 2xWrite this under x

^{2}+ 3x.3 ~ 4:

2, 4:

Bring down the next term -10.

Write it behind 5x.

Write it behind 5x.

1, 4, 5:

The goal is to remove 5x of (5x - 10)

by using x - 2.

You can make 5x

by multiplying x - 2 and 5.

So write +5 in the quotient,

on the top of +3x.

And multiply x - 2 and 5.

(x - 2)⋅(+5) = 5x - 10

Write this under 5x - 10.

by using x - 2.

You can make 5x

by multiplying x - 2 and 5.

So write +5 in the quotient,

on the top of +3x.

And multiply x - 2 and 5.

(x - 2)⋅(+5) = 5x - 10

Write this under 5x - 10.

5 ~ 6:

(5x - 10) - (5x - 10) = 0

1, 2, 6, 7:

The remainder is 0.

And the quotient is (x + 5).

So (x

So (x + 5) is the answer.

And the quotient is (x + 5).

So (x

^{2}+ 3x - 10)/(x - 2) = x + 5.So (x + 5) is the answer.