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Multiplication of Polynomials
Objective A To multiply a polynomial by a monomial
To multiply a polynomial by a monomial, use the Distribution Property and the Rule for Multiplying Exponential Expressions.
Simplify: -3a(4a 2 – 5a – 6) = -3a (4a 2 ) – (-3a * 5a ) - (-3a * 6) = -12a 3 – (-15a 2 ) – (-18a) = -12a 3 + 15a^2 + 18a
Example 1
Simplify: (5x + 4)(-2x)
Solution: (5x + 4)(-2x) = 5x(-2x) + 4(-2x) = –10x^2 – 8x
Example 2
Simplify: 2a^2 b(4a 2 – 2ab + b 2 )
Solution: 2a 2 b(4a 2 – 2ab + b 2 ) = 2a 2 b(4a 2 ) – 2a 2 b(2ab) + 2a 2 b(b2^ ) = 8a 4 b – 4a 3 b2^ + 2a 2 b
Objective B To multiply two polynomials
Multiplication of two polynomials requires the repeated application of the Distributive Property.
(y – 2)(y 2 + 3y + 1) = (y – 2)(y 2 ) + (y – 2)(3y) + (y – 2)(1) = y 3 – 2y2^ + 3y 2 – 6y + y - 2 = y 3 + y 2 – 5y - 2
A convenient method of multiplying two polynomials is to use a vertical format similar to that used for multiplication of whole numbers.
Example of using the vertical format to perform the multiplication of polynomials
y2^ + 3y + y – 2 -2y 2 – 6y –2 [Multiply y 2 +3y + 1 by –2] y3^ + 3y 2 + y. [Multiply y 2 +3y + 1 by y] y3^ + y 2 - 5y – 2 [Add the terms in each column]
The resulting answer is the same under both formats.
Simplify: ( 2a^3 + a – 3 ) ( a + 5 )
2a 3 + 0a 2 + a – 3 a + 5 10a 3 + 0a 2 + 5a - 15 2a 4 + 0a 3 + a 2 – 3a 2a 4 +10a 3. + a 2 + 2a - 15
Objective C To multiply two binomials
It is frequently necessary to find the product of two binomials. The product can be found using a method called FOIL , which is based on the Distributive Property. The letters of FOIL stand for F irst, O utside, I nside, L ast.
Simplify: (2x + 3)(x + 5)
Multiply the First terms. (2x + 3)(x + 5) 2x * x = 2x 2
Multiply the Outer terms. (2x + 3)(x + 5) 2x * 5 = 10x
Multiply the Inner terms. (2x + 3)(x + 5) 3 * x = 3x
Multiply the Last terms. (2x + 3)(x + 5) 3 * 5 = 15
Add the combined terms.
F O I L (2x + 3)(x + 5) = 2x 2 + 10x + 3x + 15 = 2x 2 + 13x + 15
Objective E: To solve application problems
Example 3
The length of a rectangle is (x + 7) m. The width is (x – 4) m. Find the area of the rectangle in terms of variable x.
x – 4
x + 7
Strategy
To find the area, replace the variable L and W in the equation A = L * W by the given values and solve for A
Solution
A = L * W
A = (x + 7)(x –4) A = x 2 – 4x + 7x – A = x 2 + 3x –
The area is (x^2 + 3x – 28) m^2.
