Multiplication of polynomials
Multiplication of two polynomials is defined as the product of each term of one polynomial by each term of the other polynomial and adding the like terms.
Multiplication of polynomials is classified as
- Multiplication of monomial by polynomial
- Multiplication of binomial by binomial
- Multiplication of polynomial by polynomial
Multiplication of Monomial by Polynomial
Multiply monomial with each term of the polynomial
Solved Problems
1) Multiply 5x by -4x + 3
Solution:
(5x)(-4x+3)
= (5x)(-4x)+(5x)(3)
= -20x2+15x
2) Multiply -7x2 by 2x2 +3x-1
Solution:
(-7x2)(2x2 +3x-1)
= (-7x2)(2x2 )+(-7x2)(3x)+(-7x2)(-1)
= -14x4-21x3+7x2
3) Multiply 2x3 by -3x3-x2 +2x-3
Solution:
(2x3)(-3x3-x2 +2x-3)
= (2x3)(-3x3)+(2x3)(-x2)+(2x3)(2x)+(2x3)(-3)
= -6x6-2x5+4x4-6x3
Multiplication of Binomial by Binomial
Multiplication of two binomialsis done by FOIL which means F(firsts) O(outers) I(Inners) L(lasts), Learn factoring polynomials.
Some standard formulas using FOIL are
1) a2 - b2 = (a+b)(a-b)
2) (a+b)2 = a2+2ab+b2
3) (a-b)2 = a2-2ab+b2
Some Problems
1) Multiply (7x+1)2
Solution: Here a = 7x and b =1 by the formula (2)
(7x+1)2 = (7x)2+2*7x*1+12
= 49x2+14x+1
2) Solve (4x+3)(4x-3)
Solution: Here a = 4x and b =3 by the formula (1)
(4x+3)(4x-3) = (4x)2-32
= 16x2-9
3) Mutliply (3x-2)2
Solution: Here a = 3x and b =2 by the formula (3)
(3x-2)2 = (3x)2-2*3x*2+22
= 9x2-12x+4
4) Multiply (7x+2)(5x+3)
Solution:
(7x+2)(5x+3) = (7x)(5x) (firsts) + (7x)(3) (outers) + (2)(5x) (inners) +(2)(3) (lasts)
= 35x2+21x+10x+6
= 35x2+31x+6
5) Multiply (-2x+1)(x-3)
Solution:
(-2x+1)(x-3) = (-2x)(x) (firsts) + (-2x)(-3) (outers) + (1)(x) (inners) +(1)(-3) (lasts)
= -2x2+6x+x-3
= -2x^2+7x-3
Multiplication of Polynomials by Polynomials
Multiplication of polynomials by polynomials is carried by mutliplying each term of one polynomial to each term of the other polynomial and add like terms
Solved Problems
1) Multiply (2x2 – 4x – 7)(2x + 1)
Solution:
2x2 – 4x – 7)(2x + 1)
= (2x2 – 4x – 7)(2x) + (2x2 – 4x – 7)(1)
= 2x2(2x) – 4x(2x) – 7(2x) + 2x2(1) – 4x(1) – 7(1)
= 4x3 – 8x2 – 14x + 2x2 – 4x – 7
= 4x3 – 8x2 + 2x2 – 14x – 4x – 7
= 4x3 - 6x2 – 18x – 7
2) Multiply (2x2 + 1)(3x2 - 3x + 3)
Solution:
(2x2 + 1)(3x2 - 3x + 3)
= 2x2(3x2 -3x + 3) + 1(3x2 -3x + 3)
= 2x2(3x2) + 2x2(-3x) + 2x2(3) +1(3x2) +1(-3x) +1(3)
= 6x4 - 6x3 + 6x2 + 3x2 - 3x + 3
= 6x4 - 6x3 + 9x2 - 3x + 3
3) Multiply (x2 + 2x + 3)(2x2 - 3x + 2)
Solution:
(x2 + 2x + 3)(2x2 - 3x + 2)
= x2(2x2 - 3x + 2)+2x(2x2 - 3x + 2)+3(2x2 - 3x + 2)
= x2(2x2) + x2(- 3x) + x2(2) + 2x(2x2) + 2x(-3x) + 2x(2) + 3(2x2) + 3(-3x) + 3(2)
= 2x4 -3x3 + 2x2 +4x3 -6x2 + 4x + 6x2 - 9x + 6
= 2x4 + (- 3 + 4)x3 + (2 - 6 + 6)x2 + (4 - 9)x + 6
= 2x4 + x3 + 2x2 - 5x + 6