Polynomials

Adding and Subtracting Polynomials

EXAMPLE

Add or subtract the following polynomials.

(a) (8x - 4x + 6x) + (3x + 5x - 9x + 8)

Solution

Combine like terms.

(8x - 4x + 6x) + (3x + 5x - 9x + 8)

= (8x + 3x) + (- 4x + 5x) + (6x - 9x) + 8

= 11x + x -3x + 8

(b) (-4x + 6x - 9x - 12) + (-3x + 8x - 11x + 7)

Solution

Combining like terms as before yields .

-4x + 3x - x - 11x - 5

(c) (2x - 11x + 8) - (7x - 6x + 2)

Solution

Distributing the minus sign yields

(2x - 11x + 8) + (-7x + 6x - 2)

= -5x - 5x + 6

Multiplying Polynomials

The distributive property is also used to multiply polynomials, along with the fact that For example,

EXAMPLE

Multiply.

(a) 8x(6x - 4)

Solution

8x(6x - 4) = 8x(6x) - 8x(4)

= 48x - 32x

(b) (3p - 2)(p + 5p - 1)

Solution

(3p - 2)(p + 5p - 1)

= 3p(p + 5p - 1) -2(p + 5p - 1)

= 3p(p) + 3p(5p) + 3p(-1) -2(p) -2(5p) -2(-1)

= 3p + 15p -3p -2p - 10p + 2

= 3p + 13p -13p + 2

(c) (x + 2)(x + 3)(x - 4)

Solution

(x + 2)(x + 3)(x - 4)

= [(x + 2)(x + 3)](x - 4)

= (x + 2x + 3x + 6)(x - 4)

= (x + 5x + 6)(x - 4)

= x + 5x + 6x -4x - 20x - 24

= x + x - 14x - 24