CPM Homework Help : CCA Problem 3-70

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3-70.

Use a generic rectangle to multiply the following expressions. Write each solution both as a sum and as a product.

$(2x+5)(x+6)$
Hint 1 (a):
Draw a generic rectangle and label the length and width.
Generic rectangle
top left edge $2x$
top right edge $+5$
left edge top $x$
interior top left blank
interior top right blank
left edge bottom $+6$
interior
bottom left blank
interior
bottom
right blank
Hint 2 (a):
Fill in the grid by multiplying the length and width of each box.
Generic rectangle
top left edge $2x$
top right edge $+5$
left edge top $x$
interior top left $2x^2$
interior top right $5x$
left edge bottom $+6$
interior bottom left $12x$
interior bottom
right $30$
Answer (a):
$(2x+5)(x+6)=2x^2+17x+30$

Write an equation showing the product is the sum of the parts.
Area as a sum: $2x^2+12x+5x+30=2x^2+17x+30$
Area as a product: $(2x+5)(x+6)$

$(m-3)(3m+5)$
Hint (b):
See the help for part (a).
Answer (b):
Generic rectangle
top left edge $m$
top right edge $-3$
left edge top $3m$
interior top left $3m^2$
interior top right $-9m$
left edge bottom $+5$
interior
bottom left $5m$
interior
bottom
right $-15$
Area as product: $(m−3)(3m+5)$
Area as sum: $3m^2−4m−15$

$(12x+1)(x^2-5)$

Hint (c):

See the help for part (a).

Partial Answer (c):

Generic rectangle

top left edge $\color{red}{12x}$

top right edge $\color{red}{+1}$

left edge top $\color{red}{x^2}$

interior top left $\color{red}{12x^3}$

interior top right $\color{red}{x^2}$

left edge bottom $\color{red}{-5}$

interior
bottom left $\color{red}{-60x}$

interior
bottom
right $\color{red}{-5}$

$(3-5y)(2+y)$
Hint (d):
See the help for part (a).

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