How to use special polynomial forms to factor other polynomials? Questions are presented along with detailed Solutions and explanations. We will study five special polynomial forms.
a 2 - b 2 = (a - b)(a + b)
Question
Factor the polynomial.
16 x 2 - 9 y 2
Solution
Note that 16 x 2 = (4 x) 2 and 9 y 2 = (3 y) 2
We can write
16 x 2 - 9 y 2 = (4 x) 2 - (3 y) 2
Now that we have written the given polynomial as the the difference of two squares, we use formula above to factor the given polynomial as follows:
16 x 2 - 9 y 2 = (4 x) 2 - (3 y) 2 = (4 x - 3 y)(4 x + 3 y)
a) a 2 + 2 a b + b 2 = (a + b) 2
b) a 2 - 2 a b + b 2 = (a - b) 2
Question
Factor the polynomials.
4 x 2 + 20 x y + 25 y 2
Solution
Note that the monomials making the given polynomial may be written as follows:
4 x 2 = (2 x) 2 , 20 x y = 2(2 x)(5 y) and 25 y 2 = (5 y) 2.
We now write the given polynomial as follows
4 x 2 + 10 x y + 25 y 2 = (2 x) 2 + 2(2 x)(5 y) + (5 y) 2
Use the formula a 2 + 2 a b + b 2 = (a + b) 2 to write the given polynomial as a square as follows:
4 x 2 + 20 x y + 25 y 2 = (2 x) 2 + 2(2 x)(5 y) + (5 y) 2 = (2 x + 5 y) 2
Question
Factor the polynomials.
1 - 6 x + 9 x 2
Solution
Note that the monomials making the given polynomial may be written as follows:
1 = 1 2 , - 6 x = - 2(3)x and 9 x 2 = (3 x) 2.
The given polynomial may be written as follows
1 - 6 x + 9 x 2 = 1 2 - 2(3) x + (3 x) 2
Use the formula a 2 - 2 a b + b 2 = (a - b) 2 to write the given polynomial as a square as follows:
1 - 6 x + 9 x 2 = 1 2 - 2(3) x + (3 x) 2 = (1 - 3 x) 2
a 3 - b 3 = (a - b)(a 2 + a b + b 2)
Question
Factor the polynomial.
8 - 27 x 3
Solution
Note that the monomials making the given polynomial may be written as follows:
8 = (2) 3 and 27 x 3 = (3 x) 3
The given polynomial may now be written as follows
8 - 27 x 3 = (2) 3 - (3 x) 3
Use the formula a 3 - b 3 = (a - b)(a 2 + ab + b 2) to write the given polynomial in factored as follows:
8 - 27 x 3 = (2) 3 - (3 x) 3 = (2 - 3 x)( (2) 2 + (2)(3x) + (3 x) 2) = (2 - 3 x)(9 x 2 + 6x + 4)
a 3 + b 3 = (a + b)(a 2 - a b + b 2)
Question
Factor the polynomial.
8 y 3 + 1
Solution
The two monomials making the given polynomial may be written as follows:
8 y 3 = (2 y) 3 and 1 = (1) 3
The polynomial to factor may now be written as follows
8 y 3 + 1 = (2 y) 3 + (1) 3
Use the formula a 3 + b 3 = (a + b)(a 2 - ab + b 2) to write the given polynomial in factored as follows:
8 y 3 + 1 = (2 y) 3 + (1) 3 = (2 y + 1)( (2 y) 2 - (2 y)(1) + (1) 2) = (2 y + 1)(4 y 2 - 2 y + 1)
Factor the following special polynomials
a) - 25 x 2 + 9
b) 16 y 4 - x 4
c) 36 y 2 - 60 x y + 25 x 2
d) (1/2) x 2 + x + (1/2)
e) - y 3 - 64
f) x 6 - 1
a) If we let a = 5 x and b = 3, the given polynomial may be written as:
- 25 x 2 + 9 = - a 2 + b 2
Use the special polynomial a 2 - b 2 = (a - b)(a + b) and factor the given polynomial as follows:
- 25 x 2 + 9 = - a 2 + b 2 = (- a + b)(a + b) = (-5 x + 3)(5 x + 3)
b) The given polynomial has the form of the difference of two squares and may be written as:
16 y 4 - x 4 = (4 y 2) 2 - (x 2) 2
Use the special polynomial a 2 - b 2 = (a - b)(a + b) and factor the given polynomial as follows:
16 y 4 - x 4 = (4 y 2) 2 - (x 2) 2 = (4y 2 - x 2)(4y 2 + x 2)
The term (4y 2 + x 2) in the above is the sum of two squares and cannot be factored using real numbers. However the term (4y 2 - x 2) is the difference of two squares and can be further factored. Hence the given polynomial is factored as follows:
16 y 4 - x 4 = (2 y - x)(2 y + x)(4y 2 + x 2)
c) The given polynomial may be written as:
36 y 2 - 60 x y + 25 x 2 = (6 y) 2 - 2(6 y)(5 x) + (5 x) 2
Use the special trinomial a 2 - 2 a b + b 2 = (a - b) 2 to factor the given polynomial as follows:
36 y 2 - 60 x y + 25 x 2 = (6 y) 2 - 2(6 y)(5 x) + (5 x) 2 = (6 y - 5 x) 2
d) Factor (1/2) out and rewrite the given polynomial as:
(1/2) x 2 + x + (1/2) = (1/2) x 2 + 2 (1/2) x + (1/2) = (1/2)( x 2 + 2 x + 1)
Use the special trinomial a 2 + 2 a b + b 2 = (a + b) 2 to factor x 2 + 2 x + 1 = x 2 + 2(x)(1) + 1 2 and the given polynomial as follows:
(1/2) x 2 + x + (1/2) = (1/2)( x 2 + 2 x + 1) = (1/2)(x + 1) 2
e) Factor - 1 out and rewrite the given polynomial as:
- y 3 - 64 = - (y 3 + 64) = - ( y 3 + 4 3)
Use a 3 + b 3 = (a + b)(a 2 - a b + b 2) to factor the given polynomial as follows:
- y 3 - 64 = - (y 3 + 64) = - ( y 3 + 4 3)
= -(y + 4)(y 2 - (y)(4) + 4 2) = -(y + 4)(y 2 - 4 y + 16)
f) Let us write the given polynomial as the difference of two squares as follows:
x 6 - 1 = (x 3) 2 - (1) 2
Use the special difference of squares polynomial a 2 - b 2 = (a - b)(a + b) and factor the given polynomial as follows:
x 6 - 1 = (x 3) 2 - (1) 2 = (x 3 - 1)(x 3 + 1)
In the above we have the product of the sum and difference of two cubes. Hence
x 6 - 1 = (x 3) 2 - (1) 2 = (x 3 - 1)(x 3 + 1)
= (x - 1)(x 2 + x + 1)(x + 1)(x 2 - x + 1)
Factor Polynomials
Factor Polynomials by Common Factor
Factor polynomial by Grouping
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