Polynomials: Multiplication
Joseph Lee
Metropolitan Community College
Joseph Lee
Polynomials: Multiplication
Example 1.a.
Multiply.
3x(5x 2 )
Joseph Lee
Polynomials: Multiplication
Example 1.a.
Multiply.
3x(5x 2 )
Solution.
3x(5x 2 ) =
Joseph Lee
Polynomials: Multiplication
Example 1.a.
Multiply.
3x(5x 2 )
Solution.
3x(5x 2 ) = 15x 3
Joseph Lee
Polynomials: Multiplication
Example 1.b.
Multiply.
−4x 3 (6x 2 )
Joseph Lee
Polynomials: Multiplication
Example 1.b.
Multiply.
−4x 3 (6x 2 )
Solution.
−4x 3 (6x 2 ) =
Joseph Lee
Polynomials: Multiplication
Example 1.b.
Multiply.
−4x 3 (6x 2 )
Solution.
−4x 3 (6x 2 ) = − 24x 5
Joseph Lee
Polynomials: Multiplication
Definition: Distributive Property
a(b + c) = ab + ac
Joseph Lee
Polynomials: Multiplication
Example 2.a.
Multiply.
3x(x + 7)
Joseph Lee
Polynomials: Multiplication
Example 2.a.
Multiply.
3x(x + 7)
Solution.
3x(x + 7) =
Joseph Lee
Polynomials: Multiplication
Example 2.a.
Multiply.
3x(x + 7)
Solution.
3x(x + 7) = 3x(x) + 3x(7)
=
Joseph Lee
Polynomials: Multiplication
Example 2.a.
Multiply.
3x(x + 7)
Solution.
3x(x + 7) = 3x(x) + 3x(7)
= 3x 2 + 21x
Joseph Lee
Polynomials: Multiplication
Example 2.b.
Multiply.
−2x 2 (3x − 5)
Joseph Lee
Polynomials: Multiplication
Example 2.b.
Multiply.
−2x 2 (3x − 5)
Solution.
−2x 2 (3x − 5) =
Joseph Lee
Polynomials: Multiplication
Example 2.b.
Multiply.
−2x 2 (3x − 5)
Solution.
−2x 2 (3x − 5) = − 2x 2 (3x) − 2x 2 (−5)
=
Joseph Lee
Polynomials: Multiplication
Example 2.b.
Multiply.
−2x 2 (3x − 5)
Solution.
−2x 2 (3x − 5) = − 2x 2 (3x) − 2x 2 (−5)
= − 6x 3 + 10x 2
Joseph Lee
Polynomials: Multiplication
Example 2.c.
Multiply.
4x(x 2 − 3x + 4)
Joseph Lee
Polynomials: Multiplication
Example 2.c.
Multiply.
4x(x 2 − 3x + 4)
Solution.
4x(x 2 − 3x + 4) =
Joseph Lee
Polynomials: Multiplication
Example 2.c.
Multiply.
4x(x 2 − 3x + 4)
Solution.
4x(x 2 − 3x + 4) = 4x(x 2 ) + 4x(−3x) + 4x(4)
=
Joseph Lee
Polynomials: Multiplication
Example 2.c.
Multiply.
4x(x 2 − 3x + 4)
Solution.
4x(x 2 − 3x + 4) = 4x(x 2 ) + 4x(−3x) + 4x(4)
= 4x 3 − 12x 2 + 16x
Joseph Lee
Polynomials: Multiplication
Example 2.d.
Multiply.
−8x 3 (−3x 2 + 7x − 1)
Joseph Lee
Polynomials: Multiplication
Example 2.d.
Multiply.
−8x 3 (−3x 2 + 7x − 1)
Solution.
−8x 3 (−3x 2 + 7x − 1) =
Joseph Lee
Polynomials: Multiplication
Example 2.d.
Multiply.
−8x 3 (−3x 2 + 7x − 1)
Solution.
−8x 3 (−3x 2 + 7x − 1) = − 8x 3 (−3x 2 ) − 8x 3 (7x) − 8x 3 (−1)
=
Joseph Lee
Polynomials: Multiplication
Example 2.d.
Multiply.
−8x 3 (−3x 2 + 7x − 1)
Solution.
−8x 3 (−3x 2 + 7x − 1) = − 8x 3 (−3x 2 ) − 8x 3 (7x) − 8x 3 (−1)
= 24x 5 − 56x 4 + 8x 3
Joseph Lee
Polynomials: Multiplication
Example 2.e.
Multiply.
(3x 2 − x + 8)(2x)
Joseph Lee
Polynomials: Multiplication
Example 2.e.
Multiply.
(3x 2 − x + 8)(2x)
Solution.
(3x 2 − x + 8)(2x) =
Joseph Lee
Polynomials: Multiplication
Example 2.e.
Multiply.
(3x 2 − x + 8)(2x)
Solution.
(3x 2 − x + 8)(2x) = 3x 2 (2x) − x(2x) + 8(2x)
=
Joseph Lee
Polynomials: Multiplication
Example 2.e.
Multiply.
(3x 2 − x + 8)(2x)
Solution.
(3x 2 − x + 8)(2x) = 3x 2 (2x) − x(2x) + 8(2x)
= 6x 3 − 2x 2 + 16x
Joseph Lee
Polynomials: Multiplication
Example 3.a.
Multiply.
(x − 2)(x + 3)
Joseph Lee
Polynomials: Multiplication
Example 3.a.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) =
Joseph Lee
Polynomials: Multiplication
Example 3.a.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) = (x − 2)(x) + (x − 2)(3)
=
Joseph Lee
Polynomials: Multiplication
Example 3.a.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) = (x − 2)(x) + (x − 2)(3)
= x 2 − 2x + 3x − 6
=
Joseph Lee
Polynomials: Multiplication
Example 3.a.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) = (x − 2)(x) + (x − 2)(3)
= x 2 − 2x + 3x − 6
= x2 + x − 6
Joseph Lee
Polynomials: Multiplication
Definition: FOIL
FOIL is a procedure for multiplying two binomials. FOIL stands for
First - Outer - Inner - Last.
(a + b)(c + d) = ac + ad + bc + bd
Joseph Lee
Polynomials: Multiplication
Example 3.b.
Multiply.
(x − 2)(x + 3)
Joseph Lee
Polynomials: Multiplication
Example 3.b.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) =
Joseph Lee
Polynomials: Multiplication
Example 3.b.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) = x 2 + 3x − 2x − 6
=
Joseph Lee
Polynomials: Multiplication
Example 3.b.
Multiply.
(x − 2)(x + 3)
Solution.
(x − 2)(x + 3) = x 2 + 3x − 2x − 6
= x2 + x − 6
Joseph Lee
Polynomials: Multiplication
Example 4.a.
Multiply.
(2x − 3)(x + 4)
Joseph Lee
Polynomials: Multiplication
Example 4.a.
Multiply.
(2x − 3)(x + 4)
Solution.
(2x − 3)(x + 4) =
Joseph Lee
Polynomials: Multiplication
Example 4.a.
Multiply.
(2x − 3)(x + 4)
Solution.
(2x − 3)(x + 4) = 2x 2 + 8x − 3x − 12
=
Joseph Lee
Polynomials: Multiplication
Example 4.a.
Multiply.
(2x − 3)(x + 4)
Solution.
(2x − 3)(x + 4) = 2x 2 + 8x − 3x − 12
= 2x 2 + 5x − 12
Joseph Lee
Polynomials: Multiplication
Example 4.b.
Multiply.
(4x − 3)(3x − 7)
Joseph Lee
Polynomials: Multiplication
Example 4.b.
Multiply.
(4x − 3)(3x − 7)
Solution.
(4x − 3)(3x − 7) =
Joseph Lee
Polynomials: Multiplication
Example 4.b.
Multiply.
(4x − 3)(3x − 7)
Solution.
(4x − 3)(3x − 7) = 12x 2 − 28x − 9x + 21
=
Joseph Lee
Polynomials: Multiplication
Example 4.b.
Multiply.
(4x − 3)(3x − 7)
Solution.
(4x − 3)(3x − 7) = 12x 2 − 28x − 9x + 21
= 12x 2 − 37x + 21
Joseph Lee
Polynomials: Multiplication
Example 4.c.
Multiply.
(x + 4)(y + 5)
Joseph Lee
Polynomials: Multiplication
Example 4.c.
Multiply.
(x + 4)(y + 5)
Solution.
(x + 4)(y + 5) =
Joseph Lee
Polynomials: Multiplication
Example 4.c.
Multiply.
(x + 4)(y + 5)
Solution.
(x + 4)(y + 5) = xy + 5x + 4y + 20
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(x − 3y )(2x + 5y )
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(x − 3y )(2x + 5y )
Solution.
(x − 3y )(2x + 5y ) =
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(x − 3y )(2x + 5y )
Solution.
(x − 3y )(2x + 5y ) = 2x 2 + 5xy − 6xy − 15y 2
=
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(x − 3y )(2x + 5y )
Solution.
(x − 3y )(2x + 5y ) = 2x 2 + 5xy − 6xy − 15y 2
= 2x 2 − xy − 15y 2
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(7x − y )(3x + y )
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(7x − y )(3x + y )
Solution.
(7x − y )(3x + y ) =
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(7x − y )(3x + y )
Solution.
(7x − y )(3x + y ) = 21x 2 + 7xy − 3xy − y 2
=
Joseph Lee
Polynomials: Multiplication
Example 4.d.
Multiply.
(7x − y )(3x + y )
Solution.
(7x − y )(3x + y ) = 21x 2 + 7xy − 3xy − y 2
= 21x 2 + 4xy − y 2
Joseph Lee
Polynomials: Multiplication
Example 4.e.
Multiply.
1
x−
2
1
x−
3
Joseph Lee
Polynomials: Multiplication
Example 4.e.
Multiply.
1
x−
2
1
x−
3
Solution.
1
x−
2
1
x−
=
3
Joseph Lee
Polynomials: Multiplication
Example 4.e.
Multiply.
1
x−
2
1
x−
3
Solution.
1
x−
2
1
1
1
1
x−
= x2 − x − x +
3
2
6
3
=
Joseph Lee
Polynomials: Multiplication
Example 4.e.
Multiply.
1
x−
2
1
x−
3
Solution.
1
x−
2
1
1
1
1
x−
= x2 − x − x +
3
2
6
3
2
3
1
= x2 − x − x +
6
6
6
=
Joseph Lee
Polynomials: Multiplication
Example 4.e.
Multiply.
1
x−
2
1
x−
3
Solution.
1
x−
2
1
1
1
1
x−
= x2 − x − x +
3
2
6
3
2
3
1
= x2 − x − x +
6
6
6
5
1
= x2 − x +
6
6
Joseph Lee
Polynomials: Multiplication
Example 5.a.
Multiply.
(x + 3)2
Joseph Lee
Polynomials: Multiplication
Example 5.a.
Multiply.
(x + 3)2
Solution.
(x + 3)2 =
Joseph Lee
Polynomials: Multiplication
Example 5.a.
Multiply.
(x + 3)2
Solution.
(x + 3)2 = (x + 3)(x + 3)
=
Joseph Lee
Polynomials: Multiplication
Example 5.a.
Multiply.
(x + 3)2
Solution.
(x + 3)2 = (x + 3)(x + 3)
= x 2 + 3x + 3x + 9
=
Joseph Lee
Polynomials: Multiplication
Example 5.a.
Multiply.
(x + 3)2
Solution.
(x + 3)2 = (x + 3)(x + 3)
= x 2 + 3x + 3x + 9
= x 2 + 6x + 9
Joseph Lee
Polynomials: Multiplication
Example 5.b.
Multiply.
(3x − 2)2
Joseph Lee
Polynomials: Multiplication
Example 5.b.
Multiply.
(3x − 2)2
Solution.
(3x − 2)2 =
Joseph Lee
Polynomials: Multiplication
Example 5.b.
Multiply.
(3x − 2)2
Solution.
(3x − 2)2 = (3x − 2)(3x − 2)
=
Joseph Lee
Polynomials: Multiplication
Example 5.b.
Multiply.
(3x − 2)2
Solution.
(3x − 2)2 = (3x − 2)(3x − 2)
= 9x 2 − 6x − 6x + 4
=
Joseph Lee
Polynomials: Multiplication
Example 5.b.
Multiply.
(3x − 2)2
Solution.
(3x − 2)2 = (3x − 2)(3x − 2)
= 9x 2 − 6x − 6x + 4
= 9x 2 − 12x + 4
Joseph Lee
Polynomials: Multiplication
Example 5.c.
Multiply.
(4x + 5y )2
Joseph Lee
Polynomials: Multiplication
Example 5.c.
Multiply.
(4x + 5y )2
Solution.
(4x + 5y )2 =
Joseph Lee
Polynomials: Multiplication
Example 5.c.
Multiply.
(4x + 5y )2
Solution.
(4x + 5y )2 = (4x + 5y )(4x + 5y )
=
Joseph Lee
Polynomials: Multiplication
Example 5.c.
Multiply.
(4x + 5y )2
Solution.
(4x + 5y )2 = (4x + 5y )(4x + 5y )
= 16x 2 + 20xy + 20xy + 25y 2
=
Joseph Lee
Polynomials: Multiplication
Example 5.c.
Multiply.
(4x + 5y )2
Solution.
(4x + 5y )2 = (4x + 5y )(4x + 5y )
= 16x 2 + 20xy + 20xy + 25y 2
= 16x 2 + 40xy + 25y 2
Joseph Lee
Polynomials: Multiplication
Example 6.a.
Multiply.
(a + b)2
Joseph Lee
Polynomials: Multiplication
Example 6.a.
Multiply.
(a + b)2
Solution.
(a + b)2 =
Joseph Lee
Polynomials: Multiplication
Example 6.a.
Multiply.
(a + b)2
Solution.
(a + b)2 = (a + b)(a + b)
=
Joseph Lee
Polynomials: Multiplication
Example 6.a.
Multiply.
(a + b)2
Solution.
(a + b)2 = (a + b)(a + b)
= a2 + ab + ab + b 2
=
Joseph Lee
Polynomials: Multiplication
Example 6.a.
Multiply.
(a + b)2
Solution.
(a + b)2 = (a + b)(a + b)
= a2 + ab + ab + b 2
= a2 + 2ab + b 2
Joseph Lee
Polynomials: Multiplication
Example 6.b.
Multiply.
(a − b)2
Joseph Lee
Polynomials: Multiplication
Example 6.b.
Multiply.
(a − b)2
Solution.
(a − b)2 =
Joseph Lee
Polynomials: Multiplication
Example 6.b.
Multiply.
(a − b)2
Solution.
(a − b)2 = (a − b)(a − b)
=
Joseph Lee
Polynomials: Multiplication
Example 6.b.
Multiply.
(a − b)2
Solution.
(a − b)2 = (a − b)(a − b)
= a2 − ab − ab + b 2
=
Joseph Lee
Polynomials: Multiplication
Example 6.b.
Multiply.
(a − b)2
Solution.
(a − b)2 = (a − b)(a − b)
= a2 − ab − ab + b 2
= a2 − 2ab + b 2
Joseph Lee
Polynomials: Multiplication
Example 7.a.
Multiply.
(x − 5)(x + 5)
Joseph Lee
Polynomials: Multiplication
Example 7.a.
Multiply.
(x − 5)(x + 5)
Solution.
(x − 5)(x + 5) =
Joseph Lee
Polynomials: Multiplication
Example 7.a.
Multiply.
(x − 5)(x + 5)
Solution.
(x − 5)(x + 5) = x 2 + 5x − 5x − 25
=
Joseph Lee
Polynomials: Multiplication
Example 7.a.
Multiply.
(x − 5)(x + 5)
Solution.
(x − 5)(x + 5) = x 2 + 5x − 5x − 25
= x 2 − 25
Joseph Lee
Polynomials: Multiplication
Example 7.b.
Multiply.
(x + 7)(x − 7)
Joseph Lee
Polynomials: Multiplication
Example 7.b.
Multiply.
(x + 7)(x − 7)
Solution.
(x + 7)(x − 7) =
Joseph Lee
Polynomials: Multiplication
Example 7.b.
Multiply.
(x + 7)(x − 7)
Solution.
(x + 7)(x − 7) = x 2 − 7x + 7x − 49
=
Joseph Lee
Polynomials: Multiplication
Example 7.b.
Multiply.
(x + 7)(x − 7)
Solution.
(x + 7)(x − 7) = x 2 − 7x + 7x − 49
= x 2 − 49
Joseph Lee
Polynomials: Multiplication
Example 7.c.
Multiply.
(3x + 5y )(3x − 5y )
Joseph Lee
Polynomials: Multiplication
Example 7.c.
Multiply.
(3x + 5y )(3x − 5y )
Solution.
(3x + 5y )(3x − 5y ) =
Joseph Lee
Polynomials: Multiplication
Example 7.c.
Multiply.
(3x + 5y )(3x − 5y )
Solution.
(3x + 5y )(3x − 5y ) = 9x 2 − 15xy + 15xy − 25y 2
=
Joseph Lee
Polynomials: Multiplication
Example 7.c.
Multiply.
(3x + 5y )(3x − 5y )
Solution.
(3x + 5y )(3x − 5y ) = 9x 2 − 15xy + 15xy − 25y 2
= 9x 2 − 25y 2
Joseph Lee
Polynomials: Multiplication
Example 8.
Multiply.
(a + b)(a − b)
Joseph Lee
Polynomials: Multiplication
Example 8.
Multiply.
(a + b)(a − b)
Solution.
(a + b)(a − b) =
Joseph Lee
Polynomials: Multiplication
Example 8.
Multiply.
(a + b)(a − b)
Solution.
(a + b)(a − b) = a2 − ab + ab − b 2
=
Joseph Lee
Polynomials: Multiplication
Example 8.
Multiply.
(a + b)(a − b)
Solution.
(a + b)(a − b) = a2 − ab + ab − b 2
= a2 − b 2
Joseph Lee
Polynomials: Multiplication
Example 9.a.
Multiply.
(x + 2)(x 2 − 3x + 2)
Joseph Lee
Polynomials: Multiplication
Example 9.a.
Multiply.
(x + 2)(x 2 − 3x + 2)
Solution.
(x + 2)(x 2 − 3x + 2) =
Joseph Lee
Polynomials: Multiplication
Example 9.a.
Multiply.
(x + 2)(x 2 − 3x + 2)
Solution.
(x + 2)(x 2 − 3x + 2) = (x + 2)(x 2 ) + (x + 2)(−3x) + (x + 2)(2)
=
Joseph Lee
Polynomials: Multiplication
Example 9.a.
Multiply.
(x + 2)(x 2 − 3x + 2)
Solution.
(x + 2)(x 2 − 3x + 2) = (x + 2)(x 2 ) + (x + 2)(−3x) + (x + 2)(2)
= x 3 + 2x 2 − 3x 2 − 6x + 2x + 4
=
Joseph Lee
Polynomials: Multiplication
Example 9.a.
Multiply.
(x + 2)(x 2 − 3x + 2)
Solution.
(x + 2)(x 2 − 3x + 2) = (x + 2)(x 2 ) + (x + 2)(−3x) + (x + 2)(2)
= x 3 + 2x 2 − 3x 2 − 6x + 2x + 4
= x 3 − x 2 − 4x + 4
Joseph Lee
Polynomials: Multiplication
Example 9.b.
Multiply.
(x − 4)(x 2 + 4x − 5)
Joseph Lee
Polynomials: Multiplication
Example 9.b.
Multiply.
(x − 4)(x 2 + 4x − 5)
Solution.
(x − 4)(x 2 + 4x − 5) =
Joseph Lee
Polynomials: Multiplication
Example 9.b.
Multiply.
(x − 4)(x 2 + 4x − 5)
Solution.
(x − 4)(x 2 + 4x − 5) = (x − 4)(x 2 ) + (x − 4)(4x) + (x − 4)(−5)
=
Joseph Lee
Polynomials: Multiplication
Example 9.b.
Multiply.
(x − 4)(x 2 + 4x − 5)
Solution.
(x − 4)(x 2 + 4x − 5) = (x − 4)(x 2 ) + (x − 4)(4x) + (x − 4)(−5)
= x 3 − 4x 2 + 4x 2 − 16x − 5x + 20
=
Joseph Lee
Polynomials: Multiplication
Example 9.b.
Multiply.
(x − 4)(x 2 + 4x − 5)
Solution.
(x − 4)(x 2 + 4x − 5) = (x − 4)(x 2 ) + (x − 4)(4x) + (x − 4)(−5)
= x 3 − 4x 2 + 4x 2 − 16x − 5x + 20
= x 3 − 21x + 20
Joseph Lee
Polynomials: Multiplication
Example 9.c.
Multiply.
(x 2 − 1)(x 2 − x − 6)
Joseph Lee
Polynomials: Multiplication
Example 9.c.
Multiply.
(x 2 − 1)(x 2 − x − 6)
Solution.
(x 2 − 1)(x 2 − x − 6) =
Joseph Lee
Polynomials: Multiplication
Example 9.c.
Multiply.
(x 2 − 1)(x 2 − x − 6)
Solution.
(x 2 − 1)(x 2 − x − 6) = (x 2 − 1)(x 2 ) + (x 2 − 1)(−x) + (x 2 − 1)(−6)
=
Joseph Lee
Polynomials: Multiplication
Example 9.c.
Multiply.
(x 2 − 1)(x 2 − x − 6)
Solution.
(x 2 − 1)(x 2 − x − 6) = (x 2 − 1)(x 2 ) + (x 2 − 1)(−x) + (x 2 − 1)(−6)
= x 4 − x 2 − x 3 + x − 6x 2 + 6
=
Joseph Lee
Polynomials: Multiplication
Example 9.c.
Multiply.
(x 2 − 1)(x 2 − x − 6)
Solution.
(x 2 − 1)(x 2 − x − 6) = (x 2 − 1)(x 2 ) + (x 2 − 1)(−x) + (x 2 − 1)(−6)
= x 4 − x 2 − x 3 + x − 6x 2 + 6
= x 4 − x 3 − 7x 2 + x + 6
Joseph Lee
Polynomials: Multiplication