Multiplying Special Polynomials
Multiplying Polynomials
Examples
Use the methods for multiplying binomials to multiply and simplify
the following.
1.Multiply and Simplify:
(3x + 5)(3x  5)  (2x + 3)(2x  3) Note that these are conjugates. Watch signs.
9
x^{2}  25  (4x^{2}  9) Remove parentheses carefully:
9x^{2}
 25  4x^{2} + 9
5x^{2}
 16
2. (3x + 5)^{2}  (2x + 3)^{2} Square the binomials and simplify.
Watch signs.
9x^{2} + 30x + 25  (4x^{2} + 12x + 9) Remove parentheses carefully:
9x^{2} + 30x + 25
 4x^{2}  12x  9
5x^{2} + 18x + 16
3. (2x + 3)^{3} Use the pattern in the example above then multiply a binomial times
polynomial
(2x + 3)^{3} = 8x^{3} + 36x^{2}
+ 54x + 27
Applications
1. Number Problem: Use the difference of squares to multiply 93 Ã— 87 .
93 = 90 + 3 and 87 = 90 â€“ 3
93 Ã— 87 = (90 + 3)( 90 â€“ 3) = 90^{2} â€“ 3^{2 }
90^{2} â€“ 3^{2} = 8100  9 = 8091
Check:
2. Number Problem: Write an expression for the sum of the squares of three
consecutive odd (or even) numbers. Then simplify the expression.
Let x be the first number, then x + 2 and x + 4 are the next two.
If x is odd
all are odd and if x is even all are even:
N = x^{2} + (x + 2)^{2} + (x + 4)^{2}
N = x^{2} + (x^{2} + 4x + 4) + (x^{2} + 8x + 16)
N = 3x^{2} + 12x + 20
Check:
Odd: x = 3
3^{2} + 5^{2} + 7^{2} = 9 + 25 + 49 = 83 or 3(3)^{2}
+ 12(3) + 20 = 83
Even: x = 2
2^{2} + 4^{2} + 6^{2} = 4 + 16 + 36 = 56 or 3(2)^{2} + 12(2) + 20 = 56
