Multiplication of Polynomial by Polynomial
Multiplication of Polynomial by Polynomial
To multiply a polynomial by a polynomial , multiply each term of the first polynomial by each term of the second polynomial and sum up the results.
How to multiply a monomial by another monomial – see §13 of this Guide.
Examples of Problem Solving
Example 1. Find the product:
a) $(2x+5)(3x-7) = 2x(3x-7)+5(3x-7) = 6x^2-14x+15x-35 = 6x^2+x-35$
b) $(x+y)(x-2y) = x(x-2y)+y(x-2y) = x^2-2xy+xy-2y^2 = x^2-xy-2y^2$
c) $(a^2-b)(b^2+1) = a^2 (b^2+1)-b(b^2+1) = a^2 b^2+a^2-b^3-b$
d) $(k+m)(k-m) = k(k-m)+m(k-m) = k^2-km+km-m^2 = k^2-m^2$
Example 2. Simplify the expression:
a) $(x-2y)(x+2y)+4y^2 = x(x+2y)-2y(x+2y)+4y^2 = x^2+2xy-2xy-4y^2+4y^2 = x^2$
b) $(a-b)(b-a)-2ab = a(b-a)-b(b-a)-2ab = ab-a^2-b^2+ab-2ab = -a^2-b^2$
Example 3*. Write as a polynomial:
$$(\frac{1}{4} x+\frac{1}{3} y+\frac{1}{2}z)(\frac{1}{4} x-\frac{1}{3} y+\frac{1}{2} z) = \frac{1}{12} (3x+4y+6z)\cdot \frac{1}{12} (3x-4y+6z)=$$
$$ = \frac{1}{144} (3x(3x-4y+6z)+4y(3x-4y+6z)+6z(3x-4y+6z) ) = $$
$$ = \frac{1}{144} (9x^2-12xy+18xz+12xy-16y^2+24yz+18xz-24yz+36z^2 )= $$
$$ = \frac{1}{144} (9x^2-16y^2+36z^2+36xz) = \frac{1}{16} x^2- \frac{1}{9} y^2+ \frac{1}{4} z^2+ \frac{1}{4} xz $$
Example 4. Solve the equation:
a) (x-1)(x+5) = (x+2)(x-4)
$ x^2+5x-x-5 = x^2-4x+2x-8 $
4x+2x = -8+5
6x = -3
x = -0,5
b) (2x-1)(2x+7) = (3+2x)(5+2x)
$4x^2+14x-2x-7 = 15+6x+10x+4x^2$
12x-16x = 15+7
-4x = 22
x = -5,5
c) $(y^2-y+1)(y^2+y+1) = y^4+y^2+y$
$ y^4+y^3+y^2-y^3-y^2-y+y^2+y+1 = y^4+y^2+y $
y = 1
d) $ (m-\frac{1}{5})(m+\frac{1}{5}) = m^2-m $
$m^2+ \frac{1}{5} m-\frac{1}{5} m- \frac{1}{25} = m^2-m $
$m = \frac{1}{25}$
ะด) (11+k)(44+k) = (16+k)(32+k)
$484+11k+44k+k^2 = 512+16k+32k+k^2$
55k-48k = 512-484
7k = 28
k = 4
Example 5. Prove an identity:
a) (x-3)(x+8) = (x-2)(x+7)-10
$$ (x-3)(x+8) = x^2+8x-3x-24 = x^2+5x-24 $$
$$(x-2)(x+7)-10 = x^2+7x-2x-14-10 = x^2+5x-24$$
The equality is an identity.
b) (x+5)(x+4)-(x+2)(x+7) = 6
$$ (x+5)(x+4) = x^2+4x+5x+20 = x^2+9x+20 $$
$$ (x+2)(x+7) = x^2+7x+2x+14 = x^2+9x+14 $$
$$ (x^2+9x+20)-(x^2+9x+14) = 6 $$
The equality is an identity.
