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
1:
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⋅x1 + 2
= 3x3

+x⋅(-5x)
= -5⋅x1 + 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
= x2⋅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
x

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
= 5x2 - 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]⋅x2 - 1⋅[y/y]
= 6x

+4xy3/2xy
= [+4/2]⋅[x/x]⋅y3 - 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]⋅x5 - 2
= 2x3

+3x4/3x2
= [+3/3]⋅x4 - 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
2:
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.