5.3 Add, Subtract, and Multiply Polynomials

Example 1

Add polynomials vertically and horizontally

1. 3x³ - 2x² + 4x - 6

+ x³ - 5x²_____+ 3

4x³ - 7x² + 4x -3

2. (2y³ + 7y² - 6y) + (-4y² + 3y - 9)

= 2y³ + 7y² - 4y² -6y + 3y - 9

= 2y³ + 3y² - 3y - 9

Example 2

Subtract polynomials vertically and horizontally

a. 7x³ - 6x² - 3x + 7 7x³ - 6x² - 3x + 7

- (6x³ + 3x³ - 7x + 5) -6x ³ – 3x² + 7x - 5

x ³ – 9x² + 4x + 2

b. (8x² - 5x + 11) - (12x² - 9x - 3)

= 8x² - 5x + 11 - 12x² + 9x+3

= -4x² + 4x + 14

Find the sum or difference.

1. (8t + 6 -5t²) - (2t³ - 3t² + 7)

-2t³ -2t² +8t -1

2. (4p² - 6p -6) + (8p ² - 7p + 4)

12p² - 13p -2

Example 3

Multiply polynomials vertically and horizont6ally

a. 3x^2 - x + 4
x __________x+ 2
6x^2 - 2x + 8	Multiply 3x^2 - x + 4 by 2.
_3x^3 - x^2 + 4x__	Multiply 3x^2 - x + 4 by x.
3x^3 - 5x^2 + 2x + 8	Combine like terms
b. (x - 3) (x^2+ 2x -5)
= (x^3- 3) _x^2_ + (x - 3) _2x_ - (x -3) _5_
= x^3 - 3x^2 + 2x^2 - 6x -5x +15
= x^3 - x^2 - 11x + 15

Example 4

Multiply three binomials

Multiply (x - 3)(x + 7)(x + 1) in a horizontal format.

(x - 3)(x + 7)(x + 1)

= (_x² +4x -21_)(x+1)

= (_x² + 4x - 21_)(x) + (_x² + 4x - 21_)(1)

= x³ + 4x² - 21x + x² + 4x - 21

= x³ + 5x² -17x -21

SPECIAL PRODUCT PATTERNS

Sum and Difference	Example
(a + b)(b - a) = a^2 - b^2	(x + 2)(x - 2) = __x^2 - 4__
Square of a Binomial
(a + b)^2 = a^2 + 2ab + b^2	(y + 4)^2 = y^2+ 8y +16
(a - b)^2 = a^2 - 2abo + b^2	(3p^2 - 2) ^2 = 9p^4 + 12p +4
Cube of a Binomial
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3	(x+1)^3= __x^3 + 3x^2 + 3x +1_
(a - b)^3 = a^3 - 3a^2b+ 3ab^2 - b^3	(r - 3)^3= _r^3 - 9x^2 + 27x -27_

Example 5

Use special product patterns

a. (7m - 3)(7m + 3) = (_7m_)² - _3_²

= 49m² - 9

b. (4t³ + 6)² = ( _4t³ )² + 6( 4t³ )(_6_) + _6_²

= 16t⁶ + 48t³ + 36

c. (xy - 4)³ = (_xy_)³ - 3(_xy_)² (_4_)+ 3(_xy_)(_4_)² - _4_³

= x³y³ - 12x²y² + 48xy - 64