Polynomial
Summary
- Ascending Order:Example
x0, x1, x2, x3, ... - Descending Order:Example
..., x3, x2, x1, x0 - 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= a2 + 2ab + b2
- (a - b)2= a2 - 2ab + b2
- (a + b)(a - b)= a2 - b2
- (a + b)3= a3 + 3a2b + 3ab2 + b3
- (a - b)3= a3 - 3a2b + 3ab2 - b3
- 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.
x2 + 3y2 - 9xy - 4x3
x2 + 3y2 - 9xy - 4x3
Solution
- 1x2 + 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.
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
x2 + 3y2 - 9xy - 4x3
Solution
- 1x2 + 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.
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
x2 + 3y2 - 9xy - 4x3
Solution
- 1x2 + 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)
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
x2 + 3y2 - 9xy - 4x3
Solution
- 1x2 + 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)
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
7x2 - 3x + 2x2 + 8x
Solution
- 17x2 - 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).
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.
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
3a2 + 2ab + ab2 - 6ab + 9ab2 - a2
Solution
- 13a2 + 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.
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)
x(3x2 - 5x + 8)
Solution
- 1x(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:
Example
Simplify the given expression.
(5 - a + 3b)a - 3(a2 + ab - 2)
(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).
And multiply -3 and each term of (a2 + 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= 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
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)
(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:
Example
Simplify the given expression.
(x2 + 7x - 1)(3x2 - x + 8)
(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
+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 (+).
So the middle term, (+)2ab, is also (+).
Example
Simplify the given expression.
(x + 3)2
(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 (+).
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 (-).
So the middle term, (-)2ab, is also (-).
Example
Simplify the given expression.
(x - 7)2
(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 (-).
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)
(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
(Use the above formula.)
103⋅97
Solution
- 1103⋅97
- 2= (100 + 3)(100 - 3)
- 3= 1002 - 32
- 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.
(x2 + 1)(x + 1)(x - 1)
(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:
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 (+).
So the signs of the right side terms
are all (+).
Example
Simplify the given expression.
(x + 2)3
(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 (+).
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:
(+), (-), (+), (-).
So the signs of the right side terms
alternate:
(+), (-), (+), (-).
Example
Simplify the given expression.
(x - 5)3
(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:
(+), (-), (+), (-).
So the signs of the right side terms
alternate:
(+), (-), (+), (-).
Dividing a Polynomial by a Monomial
Example
Simplify the given expression.
5x2 - 3x
x
Solution
- 15x2 - 3xx
- 2=5x2x-3xx
- 3= 5x - 3
1 ~ 2:
Divide each term by the denominator x.
2 ~ 3:
Example
Simplify the given expression.
12x2y + 4xy3
2xy
Solution
- 112x2y + 4xy32xy
- 2=12x2y2xy+4xy32xy
- 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
= [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
- 16x5 + 3x4 - 12x23x2
- 2=6x53x2+3x43x2-12x23x2
- 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
= [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.)
(Use the long division.)
x2 + 3x - 10
x - 2
Solution
- 1x+5
- 2x - 2x2 + 3x - 10
- 3x2 - 2x
- 45x - 10
- 55x - 10
- 60
- 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.
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.
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:
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 (x2 + 3x - 10)/(x - 2) = x + 5.
So (x + 5) is the answer.
And the quotient is (x + 5).
So (x2 + 3x - 10)/(x - 2) = x + 5.
So (x + 5) is the answer.