Question Find the product of $a(x) = 2x$ , $b(x) = -x + 7$ , and $c(x) = 3x - 5$ .

Studdy Solution

STEP 1

Assumptions
1. The functions are given as follows: - $a(x) = 2x$ - $b(x) = -x + 7$ - $c(x) = 3x - 5$
2. We need to find the product $a(x) \cdot b(x) \cdot c(x)$ .

STEP 2

First, we will multiply the functions $a(x)$ and $b(x)$ together.
$a(x) \cdot b(x) = (2x) \cdot (-x + 7)$

STEP 3

Distribute $2x$ across the terms in $b(x)$ .
$a(x) \cdot b(x) = 2x \cdot (-x) + 2x \cdot 7$

STEP 4

Perform the multiplication for each term.
$a(x) \cdot b(x) = -2x^2 + 14x$

STEP 5

Now we have the product of $a(x)$ and $b(x)$ . Next, we will multiply this result by $c(x)$ .
$(a(x) \cdot b(x)) \cdot c(x) = (-2x^2 + 14x) \cdot (3x - 5)$

STEP 6

Distribute each term in $-2x^2 + 14x$ across the terms in $c(x)$ .
$(a(x) \cdot b(x)) \cdot c(x) = (-2x^2 \cdot 3x) + (-2x^2 \cdot -5) + (14x \cdot 3x) + (14x \cdot -5)$

STEP 7

Perform the multiplication for each term.
$(a(x) \cdot b(x)) \cdot c(x) = -6x^3 + 10x^2 + 42x^2 - 70x$

STEP 8

Combine like terms.
$(a(x) \cdot b(x)) \cdot c(x) = -6x^3 + (10x^2 + 42x^2) - 70x$

STEP 9

Add the coefficients of the $x^2$ terms.
$(a(x) \cdot b(x)) \cdot c(x) = -6x^3 + 52x^2 - 70x$
The product $a(x) \cdot b(x) \cdot c(x)$ is $-6x^3 + 52x^2 - 70x$ .

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