Question

more work multiplying polynomials
n-gen math® algebra i
in the last lesson we saw various techniques and ways to think about multiplying polynomials, mostly two
binomials. what we are doing in these types of products is using the distributive property repeatedly.
exercise #1: find each of the following products by using the distributive property twice. the first is started for
you.
(a) \\((2x + 5)(3x + 2)\\)
\\(= 2x(3x + 2) + 5(3x + 2)\\)
(b) \\((7x + 3)(5x + 4)\\)
(c) \\((x - 4)(x + 8)\\)
(d) \\((x - 5)(x - 2)\\)
(e) \\((3x - 4)(x - 7)\\)
(f) \\((x + 9)(x - 9)\\)
we have now seen four techniques for multiplying binomials: the standard multiplication algorithm, the area
model, the use of f.o.i.l., and the use of double distribution. it is interesting to see how they compare.
exercise #2: consider the product \\((3x + 5)(2x - 9)\\). find the resulting polynomial using all four techniques.
(a) standard multiplication algorithm:
\\(\

$$\begin{array}{r} 3x + 5\\\\ \\times\\ 2x - 9\\\\ \\hline \\end{array}$$

\\)
(b) the area model:
\\(\

$$\begin{array}{cc} 3x & +5\\\\ \\cline{1-2} 2x & \\multicolumn{1}{|c|}{} & \\multicolumn{1}{|c|}{}\\\\ \\cline{1-2} -9 & \\multicolumn{1}{|c|}{} & \\multicolumn{1}{|c|}{}\\\\ \\cline{1-2} \\end{array}$$

\\)
(c) f.o.i.l.:
(d) double distribution:

Explanation:

Exercise #1

(a) Step1: Distribute terms

$=2x(3x+2)+5(3x+2)$

(a) Step2: Distribute again

$=6x^2+4x+15x+10$

(a) Step3: Combine like terms

$=6x^2+19x+10$

(b) Step1: Distribute terms

$=7x(5x+4)+3(5x+4)$

(b) Step2: Distribute again

$=35x^2+28x+15x+12$

(b) Step3: Combine like terms

$=35x^2+43x+12$

(c) Step1: Distribute terms

$=x(x+8)-4(x+8)$

(c) Step2: Distribute again

$=x^2+8x-4x-32$

(c) Step3: Combine like terms

$=x^2+4x-32$

(d) Step1: Distribute terms

$=x(x-2)-5(x-2)$

(d) Step2: Distribute again

$=x^2-2x-5x+10$

(d) Step3: Combine like terms

$=x^2-7x+10$

(e) Step1: Distribute terms

$=3x(x-7)-4(x-7)$

(e) Step2: Distribute again

$=3x^2-21x-4x+28$

(e) Step3: Combine like terms

$=3x^2-25x+28$

(f) Step1: Distribute terms

$=x(x-9)+9(x-9)$

(f) Step2: Distribute again

$=x^2-9x+9x-81$

(f) Step3: Combine like terms

$=x^2-81$

(a) Standard Multiplication Algorithm

Step1: Multiply by $-9$
$(3x+5)(-9) = -27x-45$
Step2: Multiply by $2x$
$(3x+5)(2x) = 6x^2+10x$
Step3: Sum the results
$6x^2+10x-27x-45 = 6x^2-17x-45$

(b) Area Model

Step1: Fill in the grid

	$3x$	$+5$
$-9$	$-27x$	$-45$

Step2: Sum all grid terms
$6x^2+10x-27x-45 = 6x^2-17x-45$

(c) F.O.I.L.

Step1: Multiply First terms
$3x \cdot 2x = 6x^2$
Step2: Multiply Outer terms
$3x \cdot (-9) = -27x$
Step3: Multiply Inner terms
$5 \cdot 2x = 10x$
Step4: Multiply Last terms
$5 \cdot (-9) = -45$
Step5: Sum and combine like terms
$6x^2-27x+10x-45 = 6x^2-17x-45$

(d) Double Distribution

Step1: Distribute first binomial
$3x(2x-9)+5(2x-9)$
Step2: Distribute again
$6x^2-27x+10x-45$
Step3: Combine like terms
$6x^2-17x-45$

Answer:

(a) $\boldsymbol{6x^2+19x+10}$
(b) $\boldsymbol{35x^2+43x+12}$
(c) $\boldsymbol{x^2+4x-32}$
(d) $\boldsymbol{x^2-7x+10}$
(e) $\boldsymbol{3x^2-25x+28}$
(f) $\boldsymbol{x^2-81}$

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Exercise #2