Question

use a table to multiply the polynomials. the first one has been started for you.

1. $(a - 6)(2a + 3b + 5)$

step 1 - create a table and place the terms of the two polynomials along the top and left margins.
step 2 - complete the table. multiply the first term in the left margin with each of the terms in the top margin and fill in the corresponding entry. repeat for each term in the left margin.
step 3 - write up an expression for the sum of all the products and simplify the expression.

1. $(7x + 2)(x^2 + 5x + 1)$

step 1 - create a table and place the terms of the two polynomials along the top and left margins.
step 2 - create and complete the table. multiply the first term in the left margin with each of the terms in the top margin and fill in the corresponding entry. repeat for each term in the left margin.
step 3 - write up an expression for the sum of all the products and simplify the expression.

Explanation:

Step 1: Complete first polynomial table

	$+2a$	$+3b$	$+5$
$-6$	$-12a$	$-18b$	$-30$

Step 2: Sum and simplify first polynomial

Sum all table entries:
$2a^2 + 3ab + 5a - 12a - 18b - 30$
Combine like terms:
$2a^2 + 3ab + (5a-12a) - 18b - 30 = 2a^2 + 3ab -7a -18b -30$

Step 3: Complete second polynomial table

	$x^2$	$5x$	$1$
$2$	$2x^2$	$10x$	$2$

Step 4: Sum and simplify second polynomial

Sum all table entries:
$7x^3 + 35x^2 + 7x + 2x^2 + 10x + 2$
Combine like terms:
$7x^3 + (35x^2+2x^2) + (7x+10x) + 2 = 7x^3 + 37x^2 +17x +2$

Answer:

1. $2a^2 + 3ab -7a -18b -30$
2. $7x^3 + 37x^2 +17x +2$