Polynomial
See how to simplify a polynomial.
16 examples and their solutions.
- 7x2 - 3x + 2x2 + 8x
= (7 + 2)x2 + (-3 + 8)x - 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
= axa + aya - Long Division:
f(x) ÷ (x - a)
Adding and Subtracting the Terms of a Polynomial
Example
7x2 - 3x + 2x2 + 8x
Solution 7x2 - 3x + 2x2 + 8x
= (7 + 2)x2 + (-3 + 8)x
= 9x2 + 5x
= (7 + 2)x2 + (-3 + 8)x
= 9x2 + 5x
Example
3a2 + 2ab + ab2 - 6ab + 9ab2 - a2
Solution 3a2 + 2ab + ab2 - 6ab + 9ab2 - a2
= (3 - 1)a2 + (1 + 9)ab2 + (2 - 6)ab
= 2a2 + 10ab2 - 4ab
= (3 - 1)a2 + (1 + 9)ab2 + (2 - 6)ab
= 2a2 + 10ab2 - 4ab
Multiplying a Monomial and a Polynomial
Example
x(3x2 - 5x + 8)
Solution x(3x2 - 5x + 8)
= x⋅3x2 + x⋅(-5x) + x⋅8
= 3x3 - 5x2 + 8x
= x⋅3x2 + x⋅(-5x) + x⋅8
= 3x3 - 5x2 + 8x
Example
(5 - a + 3b)a - 3(a2 + ab - 2)
Solution (5 - a + 3b)a - 3(a2 + ab - 2)
= 5⋅a - a⋅a + 3b⋅a - 3⋅a2 - 3⋅ab - 3⋅(-2)
= 5a - a2 + 3ab - 3a2 - 3ab + 6
= -4a2 + 5a + 6
= 5⋅a - a⋅a + 3b⋅a - 3⋅a2 - 3⋅ab - 3⋅(-2)
= 5a - a2 + 3ab - 3a2 - 3ab + 6
= -4a2 + 5a + 6
FOIL Method: Multiplying Two Binomials
Example
(x - 2)(x - 3)
Solution (x - 2)(x - 3)
= x⋅x + x⋅(-3) - 2⋅x - 2⋅(-3)- [1]
= x2 - 3x - 2x + 6
= x2 - 5x + 6
[1] Multiply the First two terms: x⋅x.
Multiply the Outer terms: +x⋅(-3).
Multiply the Inner two terms: -2⋅x.
Multiply the Last two terms: -2⋅(-3).
= x⋅x + x⋅(-3) - 2⋅x - 2⋅(-3)- [1]
= x2 - 3x - 2x + 6
= x2 - 5x + 6
[1] Multiply the First two terms: x⋅x.
Multiply the Outer terms: +x⋅(-3).
Multiply the Inner two terms: -2⋅x.
Multiply the Last two terms: -2⋅(-3).
Multiplying Polynomials
Example
(x - 3)(x2 - x + 7)
Solution (x - 3)(x2 - x + 7)
= x⋅x2 + x⋅(-x) + x⋅7
-3⋅x2 - 3⋅(-x) - 3⋅7
= x3 - x2 + 7x
-3x2 + 3x - 21
= x3 - 4x2 + 10x - 21
= x⋅x2 + x⋅(-x) + x⋅7
-3⋅x2 - 3⋅(-x) - 3⋅7
= x3 - x2 + 7x
-3x2 + 3x - 21
= x3 - 4x2 + 10x - 21
Example
(x2 + 7x - 1)(3x2 - x + 8)
Solution (x2 + 7x - 1)
= x2⋅3x2 + x2⋅(-x) + x2⋅8
+7x⋅3x2 + 7x⋅(-x) + 7x⋅8
-x⋅3x2 - x⋅(-x) - x⋅8
= 3x4 - x3 + 8x2
+21x3 - 7x2 + 56x
-3x2 + x - 8
= 3x4 + 20x3 - 2x2 + 57x - 8
= x2⋅3x2 + x2⋅(-x) + x2⋅8
+7x⋅3x2 + 7x⋅(-x) + 7x⋅8
-x⋅3x2 - x⋅(-x) - x⋅8
= 3x4 - x3 + 8x2
+21x3 - 7x2 + 56x
-3x2 + x - 8
= 3x4 + 20x3 - 2x2 + 57x - 8
(a + b)2
Formula
(a + b)2
= a2 + 2ab + b2
= a2 + 2ab + b2
Example
(x + 3)2
Solution (x + 3)2
= x2 + 2⋅x⋅3 + 32
= x2 + 6x + 9
= x2 + 2⋅x⋅3 + 32
= x2 + 6x + 9
(a - b)2
Formula
(a - b)2
= a2 - 2ab + b2
= a2 - 2ab + b2
Example
(x - 7)2
Solution (x - 7)2
= x2 - 2⋅x⋅7 + 72
= x2 - 14x + 49
= x2 - 2⋅x⋅7 + 72
= x2 - 14x + 49
(a + b)(a - b)
Formula
(a + b)(a - b)
= a2 - b2
= a2 - b2
Example
(x + 2)(x - 2)
Solution (x + 2)(x - 2)
= x2 - 22
= x2 - 4
= x2 - 22
= x2 - 4
Example
103⋅97
(Use the above formula.)
Solution (Use the above formula.)
103⋅97
= (100 + 3)(100 - 3)
= 1002 - 32
= 10000 - 9
= 9991
= (100 + 3)(100 - 3)
= 1002 - 32
= 10000 - 9
= 9991
Example
(x2 + 1)(x + 1)(x - 1)
Solution (x2 + 1)(x + 1)(x - 1)
= (x2 + 1)(x2 - 12)
= (x2 + 1)(x2 - 1)
= (x2)2 - 12
= x4 - 1
= (x2 + 1)(x2 - 12)
= (x2 + 1)(x2 - 1)
= (x2)2 - 12
= x4 - 1
(a + b)3
Formula
(a + b)3
= a3 + 3a2b + 3ab2 + b3
= a3 + 3a2b + 3ab2 + b3
Example
(x + 2)3
Solution (x + 2)3
= x3 + 3⋅x2⋅2 + 3⋅x⋅22 + 23
= x3 + 6x2 + 12x + 8
= x3 + 3⋅x2⋅2 + 3⋅x⋅22 + 23
= x3 + 6x2 + 12x + 8
(a - b)3
Formula
(a - b)3
= a3 - 3a2b + 3ab2 - b3
= a3 - 3a2b + 3ab2 - b3
Example
(x - 5)3
Solution (x - 5)3
= x3 - 3⋅x2⋅5 + 3⋅x⋅52 - 53
= x3 - 15x2 + 75x + 8
= x3 - 3⋅x2⋅5 + 3⋅x⋅52 - 53
= x3 - 15x2 + 75x + 8
Dividing a Polynomial by a Monomial
Example
5x2 - 3xx
Solution 5x2 - 3xx
= 5x2x - 3xx
= 5x - 3
= 5x2x - 3xx
= 5x - 3
12x2y + 4xy32xy
Solution 12x2y + 4xy32xy
= 12x2y2xy + 4xy32xy
= 6x + 2y2
= 12x2y2xy + 4xy32xy
= 6x + 2y2
6x5 + 3x4 - 12x23x2
Solution 6x5 + 3x4 - 12x23x2
= 6x53x2 + 3x43x2 - 12x23x2
= 2x3 + x2 - 4
= 6x53x2 + 3x43x2 - 12x23x2
= 2x3 + x2 - 4
Long Division
Example
x2 + 3x - 10x - 2
(Use the long division.)
Solution (Use the long division.)
x+5
x - 2x2 + 3x - 10
x2 - 2x- [1]
5x - 10
5x - 10- [2]
0
(given) = x + 5
[1] x(x - 2) = x2 - 2x
[2] +5(x - 2) = 5x - 10
x - 2x2 + 3x - 10
x2 - 2x- [1]
5x - 10
5x - 10- [2]
0
(given) = x + 5
[1] x(x - 2) = x2 - 2x
[2] +5(x - 2) = 5x - 10