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:

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

Step1: Fill table with $+a$ products

Multiply $a$ by each top term:
$a \times 2a = 2a^2$, $a \times 3b = 3ab$, $a \times 5 = 5a$

	$+2a$	$+3b$	$+5$
$-6$

Step2: Fill table with $-6$ products

Multiply $-6$ by each top term:
$-6 \times 2a = -12a$, $-6 \times 3b = -18b$, $-6 \times 5 = -30$

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

Step3: Sum and simplify terms

Combine all products and like terms:
$2a^2 + 3ab + 5a -12a -18b -30 = 2a^2 + 3ab -7a -18b -30$

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Problem 2: $(7x + 2)(x^2 + 5x + 1)$

Step1: Fill table with $7x$ products

Multiply $7x$ by each top term:
$7x \times x^2 = 7x^3$, $7x \times 5x = 35x^2$, $7x \times 1 = 7x$

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

Step2: Fill table with $2$ products

Multiply $2$ by each top term:
$2 \times x^2 = 2x^2$, $2 \times 5x = 10x$, $2 \times 1 = 2$

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

Step3: Sum and simplify terms

Combine all products and like terms:
$7x^3 + 35x^2 + 7x + 2x^2 +10x +2 = 7x^3 +37x^2 +17x +2$

Answer:

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