Polynomial

Summary

Ascending Order:
x⁰, x¹, x², x³, ...
Example
Descending Order:
..., x³, x², x¹, x⁰
Example
Adding and Subtracting Polynomials
→ add and subtract like terms.
Example
a(x + y + z)

= ax + ay + az

Example
(a + b)(x + y)

= ax + ay + bx + by

Example
(a + b)(x + y + z)

= ax + ay + az

+bx + by + bz

Example
(a + b)²

= a² + 2ab + b²

Example
(a - b)²

= a² - 2ab + b²

Example
(a + b)(a - b)

= a² - b²

Example
(a + b)³

= a³ + 3a²b + 3ab² + b³

Example
(a - b)³

= a³ - 3a²b + 3ab² - b³

Example
ax + ay
a

= x + y

Example
Long Division:
f(x) ÷ (x - a)
Example

Ascending Order

Example

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

x² + 3y² - 9xy - 4x³

Solution

1

x² + 3y² - 9xy - 4x³
2

= 3y² - 9xy + x² - 4x³

See the exponents of x.

First write the x⁰ term: 3y².
(x⁰ term: doesn't have x)
Write the x¹ term: -9xy.
Write the x² term: +x².
And write the remaining term: -4x³.

Example

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

x² + 3y² - 9xy - 4x³

Solution

1

x² + 3y² - 9xy - 4x³
2

= -4x³ + x² - 9xy + 3y²

See the exponents of y.

First write the y⁰ terms: -4x³ + x².
(y⁰ terms: don't have x)
Write the y¹ term: -9xy.
And write the remaining term: +3y².

Descending Order

Example

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

x² + 3y² - 9xy - 4x³

Solution

1

x² + 3y² - 9xy - 4x³
2

= -4x³ + x² - 9xy + 3y²

See the exponents of x.

First write the x³ term: -4x³.
Write the x² term: +x².
Write the x¹ term: -9xy.
And write the remaining x⁰ term: +3y².
(x⁰ term: doesn't have x)

Example

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

x² + 3y² - 9xy - 4x³

Solution

1

x² + 3y² - 9xy - 4x³
2

= 3y² - 9xy -4x³ + x²

See the exponents of y.

First write the y² term: 3y².
Write the y¹ term: -9xy.
And write the remaining y⁰ terms: -4x³ + x².
(y⁰ terms: don't have x)

Adding and Subtracting Polynomials

Example

Simplify the given expression.

7x² - 3x + 2x² + 8x

Solution

1

7x² - 3x + 2x² + 8x
2

= (7 + 2)x² + (-3 + 8)x
3

= 9x² + 5x

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

In this polynomial,
7x² and +2x² are like terms,
because they have the same x².

By the same way,
-3x and +8x are like terms,
because they have the same x (= x¹).

1 ~ 2:

Only like terms can be added or subtracted
by adding or subtracting their coefficients.

7x² and +2x² are like terms.
So 7x² + 2x² = (7 + 2)x².

-3x and +8x are like terms.
So -3x + 8x = (-3 + 8)x.

Example

Simplify the given expression.

3a² + 2ab + ab² - 6ab + 9ab² - a²

Solution

1

3a² + 2ab + ab² - 6ab + 9ab² - a²
2

= (3 - 1)a² + (1 + 9)ab² + (2 - 6)ab
3

= 2a² + 10ab² - 4ab

1 ~ 2:

3a² and -a² are like terms.
So 3a² - a² = (3 - 1)a².

+ab² and +9ab² are like terms.
So +ab² + 9ab² = +(1 + 9)ab².

+2ab and -6ab are like terms.
So +2ab - 6ab = +(2 - 6)ab.

ab² term is written before ab term.
There's a reason.
Usually, we write the terms in descending order.
Both ab² and ab have the same power a.
But ab² has the higher power b: b².
So you should write the ab² term
before the ab term.

Multiplying a Monomial and a Polynomial

Example

Simplify the given expression.

x(3x² - 5x + 8)

Solution

1

x(3x² - 5x + 8)
2

= x⋅3x² + x⋅(-5x) + x⋅8
3

= 3x³ - 5x² + 8x

1 ~ 2:

Multiply x and each term of (3x² - 5x + 8).

2 ~ 3:

x⋅3x²
= 3⋅x^{1 + 2}
= 3x³

+x⋅(-5x)
= -5⋅x^{1 + 1}
= -5x²

Product of Powers

Example

Simplify the given expression.

(5 - a + 3b)a - 3(a² + ab - 2)

Solution

1

(5 - a + 3b)a - 3(a² + ab - 2)
2

= 5⋅a - a⋅a + 3b⋅a - 3⋅a² - 3⋅ab - 3⋅(-2)
3

= 5a - a² + 3ab - 3a² - 3ab + 6
4

= -4a² + 5a + 6

1 ~ 2:

Multiply each term of (5 - a + 3b) and a.
And multiply -3 and each term of (a² + ab - 2).

3 ~ 4:

-a² - 3a²
= (-1 - 3)a²
= -4a²

Adding and Subtracting Polynomials

Cancel +3ab and -3ab.

FOIL Method

Example

Simplify the given expression.

(x - 2)(x - 3)

Solution

1

(x - 2)(x - 3)
2

= x⋅x + x⋅(-3) - 2⋅x - 2⋅(-3)
3

= x² - 3x - 2x + 6
4

= x² - 5x + 6

1 ~ 2:

The FOIL method is a way
to multiply two binomials.

First multiply the First two terms.
x⋅x = x²

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)(x² - x + 7)

Solution

1

(x - 3)(x² - x + 7)
2

= x⋅x² + x⋅(-x) + x⋅7
3

-3⋅x² - 3⋅(-x) - 3⋅7
4

= x³ - x² + 7x
5

-3x² + 3x - 21
6

= x³ - 4x² + 10x - 21

1, 2:

Multiply x and each term of (x² - x + 7).

1, 3:

And multiply -3 and each term of (x² - x + 7).

4 ~ 6:

-x² - 3x² = -4x²
+7x + 3x = +10x

Adding and Subtracting Polynomials

Example

Simplify the given expression.

(x² + 7x - 1)(3x² - x + 8)

Solution

1

(x² + 7x - 1)
2

= x²⋅x² + x²⋅(-x) + x²⋅8
3

+7x⋅x² + 7x⋅(-x) + 7x⋅8
4

-x⋅x² - x⋅(-x) - x⋅8
5

= 3x⁴ - x³ + 8x²
6

+21x³ - 7x² + 56x
7

-3x² + x - 8
8

= 3x⁴ + 20x³ - 2x² + 57x - 8

1, 2:

Multiply x² and each term of (3x² - x + 8).

1, 3:

Multiply +7x and each term of (3x² - x + 8).

1, 4:

And multiply -1 and each term of (3x² - x + 8).

5 ~ 8:

-x³ + 21x³ = +20x³

+8x² - 7x² - 3x²
= (+8 - 7 - 3)x²
= -2x²

+56x + x = +57x

Square of a Sum: (a + b)²

Formula

(a + b)²

= a² + 2ab + b²

The middle sign in (a + b)² is (+).

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

Example

Simplify the given expression.

(x + 3)²

Solution

1

(x + 3)²
2

= x² + 2⋅x⋅3 + 3²
3

= x² + 6x + 9

1 ~ 2:

The middle sign in (x + 3)² is (+).
So the middle term, +2⋅x⋅3,
is also (+).

Square of a Difference: (a - b)²

Formula

(a - b)²

= a² - 2ab + b²

The middle sign in (a - b)² is (+).

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

Example

Simplify the given expression.

(x - 7)²

Solution

1

(x - 7)²
2

= x² - 2⋅x⋅7 + 7²
3

= x² - 14x + 49

1 ~ 2:

The middle sign in (x - 7)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)

= a² - b²

Example

Simplify the given expression.

(x + 2)(x - 2)

Solution

1

(x + 2)(x - 2)
2

= x² - 2²
3

= x² - 4

Example

Find the value of the given expression.
(Use the above formula.)

103⋅97

Solution

1

103⋅97
2

= (100 + 3)(100 - 3)
3

= 100² - 3²
4

= 10000 - 9
5

= 9991

1 ~ 2:

103 = 100 + 3
97 = 100 - 3

2 ~ 3:

Use the formula.

Example

Simplify the given expression.

(x² + 1)(x + 1)(x - 1)

Solution

1

(x² + 1)(x + 1)(x - 1)
2

= (x² + 1)(x² - 1²)
3

= (x² + 1)(x² - 1)
4

= (x²)² - 1²
5

= x⁴ - 1

1 ~ 2:

Use the formula.

2 ~ 3:

-1² = -1

3 ~ 4:

Use the formula again.

4 ~ 5:

(x²)²
= x^2⋅2
= x⁴

Power of a Power

Cube of a Sum: (a + b)³

Formula

(a + b)³

= a³ + 3a²b + 3ab² + b³

The middle sign in (a + b)³ is (+).

So the signs of the right side terms
are all (+).

Example

Simplify the given expression.

(x + 2)³

Solution

1

(x + 2)³
2

= x³ + 3⋅x²⋅2 + 3⋅x⋅2² + 2³
3

= x³ + 6x² + 12x + 8

1 ~ 2:

The middle sign in (x + 2)³ is (+).
So the middle term, +2⋅x⋅3,
is also (+).

Cube of a Difference: (a + b)³

Formula

(a - b)³

= a³ - 3a²b + 3ab² - b³

The middle sign in (a - b)³ is (-).

So the signs of the right side terms
alternate:
(+), (-), (+), (-).

Example

Simplify the given expression.

(x - 5)³

Solution

1

(x - 5)³
2

= x³ - 3⋅x²⋅5 + 3⋅x⋅5² - 5³
3

= x³ - 15x² + 75x + 8

1 ~ 2:

The middle sign in (x - 5)³ is (-).
So the signs of the right side terms
alternate:
(+), (-), (+), (-).

Dividing a Polynomial by a Monomial

Example

Simplify the given expression.

5x² - 3x

Solution

1

5x² - 3x
x
2

=

5x²
x

-

3x
x
3

= 5x - 3

1 ~ 2:

Divide each term by the denominator x.

2 ~ 3:

5x²/x
= 5x^{2 - 1}
= 5x

Quotient of Powers

Example

Simplify the given expression.

12x²y + 4xy³

2xy

Solution

1

12x²y + 4xy³
2xy
2

=

12x²y
2xy

+

4xy³
2xy
3

= 6x + 2y²

1 ~ 2:

Divide each term by the denominator 2xy.

2 ~ 3:

12x²y/2xy
= [12/2]⋅x^{2 - 1}⋅[y/y]
= 6x

+4xy³/2xy
= [+4/2]⋅[x/x]⋅y^{3 - 1}
= +2y²

Example

Simplify the given expression.

6x⁵ + 3x⁴ - 12x²

3x²

Solution

1

6x⁵ + 3x⁴ - 12x²
3x²
2

=

6x⁵
3x²

+

3x⁴
3x²

-

12x²
3x²
3

= 2x³ + x² - 4

1 ~ 2:

Divide each term by the denominator 3x².

2 ~ 3:

6x⁵/3x²
= [6/3]⋅x^{5 - 2}
= 2x³

+3x⁴/3x²
= [+3/3]⋅x^{4 - 2}
= +x²

-12x²/3x²
= [-12/3]⋅[x²/x²]
= -4

Long Division

Example

Simplify the given expression.
(Use the long division.)

x² + 3x - 10

x - 2

Solution

1

x
+5
2

x - 2
x² + 3x - 10
3

x² - 2x
4

5x - 10
5

5x - 10
6

0
7

(given) = x + 5

Just like dividing numbers,
draw the division form like this.

Write the numerator (x² + 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 x² of (x² + 3x - 10)
by using x - 2.

You can make x²
by multiplying x - 2 and x.

So write x
on the top of x².
And multiply x - 2 and x.

(x - 2)⋅x = x² - 2x
Write this under x² + 3x.

3 ~ 4:

(x² + 3x) - (x² - 2x)
= x² + 3x - x² + 2x
= +5x

Multiply a Monomial and a Polynomial

2, 4:

Bring down the next term -10.
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.

5 ~ 6:

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

1, 2, 6, 7:

The remainder is 0.
And the quotient is (x + 5).
So (x² + 3x - 10)/(x - 2) = x + 5.

So (x + 5) is the answer.

Polynomial

Summary

Ascending Order

Example

Solution

Example

Solution

Descending Order

Example

Solution

Example

Solution

Adding and Subtracting Polynomials

Example

Solution

Example

Solution

Multiplying a Monomial and a Polynomial

Example

Solution

Example

Solution

FOIL Method

Example

Solution

Multiplying Polynomials

Example

Solution

Example

Solution

Square of a Sum: (a + b)2

Formula

Example

Solution

Square of a Difference: (a - b)2

Formula

Example

Solution

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

Formula

Example

Solution

Example

Solution

Example

Solution

Cube of a Sum: (a + b)3

Formula

Example

Solution

Cube of a Difference: (a + b)3

Formula

Example

Solution

Dividing a Polynomial by a Monomial

Example

Solution

Example

Solution

Example

Solution

Long Division

Example

Solution

Square of a Sum: (a + b)²

Square of a Difference: (a - b)²

Cube of a Sum: (a + b)³

Cube of a Difference: (a + b)³