Compound Integral Expressions.

$2 x^{3} - 3 x^{2} + x$ and $2 x^{2}$ .

$a^{2} - 5 a b - b^{2}$ and $5 a b$ .

$a^{3} + 2 a^{2} b + 2 a b^{2}$ and $a^{2}$ .

$a^{3} + 2 a^{2} b + 2 a b^{2}$ and $b^{3}$ .

$4 x^{2} - 6 x y - 9 y^{2}$ and $2 x$ .

$- x^{2} - 2 x y + y^{2}$ and $- y$ .

$- a^{3} - a^{2} b^{2} - b^{3}$ and $- a^{2}$ .

$- x^{2} + 2 x y - y^{2}$ and $- y^{2}$ .

$3 a^{2} b^{2} - 4 a b^{3} + a^{3} b$ and $5 a^{2} b^{2}$ .

$- a x^{2} + 3 a x y^{2} - a y^{4}$ and $- 3 a y^{2}$ .

$x^{12} - x^{10} y^{3} - x^{3} y^{10}$ and $x^{3} y^{2}$ .

$- 2 x^{3} + 3 x^{2} y^{2} - 2 x y^{3}$ and $- 2 x^{2} y^{3}$ .

$a^{3} x^{2} y^{5} - a^{2} x y^{4} - a y^{3}$ and $a^{7} x^{3} y^{5}$ .

$3 a^{2} b^{2} - 2 a b^{3} + 5 a^{3} b$ and $5 a^{2} b^{3}$ .

87. Multiplication. Polynomials by Polynomials.

If we have $m + n + p$ to be multiplied by $a + b + c$ , we may substitute $M$ for the multiplier $a + b + c$ . Then $M (m + n + p) = M m + M n + M p .$

If now we substitute $a + b + c$ for $M$ , we shall have

find the product of two polynomials, therefore,

Multiply every term of the multiplicand by each term of the multiplier, and add the partial products.

In multiplying polynomials, it is a convenient arrangement to write the multiplier under the multiplicand, and place like terms of the partial products in columns.

Multiply $2 x - 3 y$ by $5 x - 4 y$ . $c c r r c r 2 x & - & 3 & y & & 5 x & - & 4 & y & & \cline 1 - 4 10 x^{2} & - & 15 & x y & & & - & 8 & x y & + & 12 y^{2} \hline 10 x^{2} & - & 23 & x y & + & 12 y^{2}$

We multiply $2 x$ , the first term of the multiplicand, by $5 x$ , the first term of the multiplier, and obtain $10 x^{2}$ , then $- 3 y$ , the second term of the multiplicand, by $5 x$ , and obtain $- 15 x y$ . The first line of partial products is $10 x^{2} - 15 x y$ . In multiplying by $- 4 y$ , we obtain for a second line of partial products $- 8 x y + 12 y^{2}$ , which is put one place to the right, so that the like terms $- 15 x y$ and $- 8 x y$ may stand in the same column. We then add the coefficients of the like terms, and obtain the complete product in its simplest form.