it is critical to understand that when we multiply two polynomials then our result is equivalent to this product and this equivalence can be tested.
exercise #4: consider the product of $(x - 2)$ and $(2x - 5)$.
(a) evaluate this product for $x = 4$. show the work that leads to your result.
(b) find a trinomial that represents the product of these two binomials.
(c) evaluate the trinomial for $x = 4$. is it equivalent to the answer you found in (a)?
(d) what is the value of the trinomial when $x = 2$? can you explain why this makes sense based on the two binomials?
exercise #5: the product of three binomials, just like the product of two, can be found with repeated applications of the distributive property.
(a) find the product: $(x - 2)(x + 4)(x - 7)$. use area arrays to help keep track of the product.
(b) for what three values of $x$ will the cubic polynomial that you found in part (a) have a value of zero? what famous law is this known as?
(c) test one of the three values you found in (b) to verify that it is a zero of the cubic polynomial.

Question

Accepted Answer

### Exercise #4
#### (a)
# Explanation:
## Step1: Substitute $ x = 4 $ into the binomials
First, find the value of $ (x - 2) $ when $ x = 4 $: $ 4 - 2 = 2 $
Then, find the value of $ (2x - 5) $ when $ x = 4 $: $ 2\times4 - 5 = 8 - 5 = 3 $
## Step2: Multiply the two results
Multiply the two values: $ 2\times3 = 6 $
# Answer:
$ 6 $

#### (b)
# Explanation:
## Step1: Use the distributive property (FOIL method)
Multiply $ (x - 2)(2x - 5) $:
First: $ x\times2x = 2x^2 $
Outer: $ x\times(-5) = -5x $
Inner: $ -2\times2x = -4x $
Last: $ -2\times(-5) = 10 $
## Step2: Combine like terms
Combine the middle terms: $ -5x - 4x = -9x $
So the trinomial is $ 2x^2 - 9x + 10 $
# Answer:
$ 2x^2 - 9x + 10 $

#### (c)
# Explanation:
## Step1: Substitute $ x = 4 $ into the trinomial
Substitute $ x = 4 $ into $ 2x^2 - 9x + 10 $:
$ 2\times4^2 - 9\times4 + 10 = 2\times16 - 36 + 10 $
## Step2: Calculate the value
Calculate each term: $ 32 - 36 + 10 = 6 $
Compare with the answer in (a) (which is 6), so they are equivalent.
# Answer:
The value is $ 6 $, and it is equivalent to the answer in (a).

#### (d)
# Explanation:
## Step1: Substitute $ x = 2 $ into the trinomial
Substitute $ x = 2 $ into $ 2x^2 - 9x + 10 $:
$ 2\times2^2 - 9\times2 + 10 = 2\times4 - 18 + 10 $
## Step2: Calculate the value
Calculate each term: $ 8 - 18 + 10 = 0 $
Explanation: When $ x = 2 $, the first binomial $ (x - 2) $ becomes $ 0 $, so the product of the two binomials (which is the trinomial) should also be $ 0 $, which matches the calculation.
# Answer:
The value is $ 0 $. When $ x = 2 $, the binomial $ (x - 2) = 0 $, so the product of the two binomials (the trinomial) is $ 0 $, which makes sense.

### Exercise #5
#### (a)
# Explanation:
## Step1: First multiply two binomials
First, multiply $ (x - 2)(x + 4) $ using FOIL:
First: $ x\times x = x^2 $
Outer: $ x\times4 = 4x $
Inner: $ -2\times x = -2x $
Last: $ -2\times4 = -8 $
Combine like terms: $ 4x - 2x = 2x $, so $ (x - 2)(x + 4) = x^2 + 2x - 8 $
## Step2: Multiply the result by the third binomial
Now multiply $ (x^2 + 2x - 8)(x - 7) $
Using the distributive property:
$ x^2\times x = x^3 $
$ x^2\times(-7) = -7x^2 $
$ 2x\times x = 2x^2 $
$ 2x\times(-7) = -14x $
$ -8\times x = -8x $
$ -8\times(-7) = 56 $
## Step3: Combine like terms
Combine $ x^2 $ terms: $ -7x^2 + 2x^2 = -5x^2 $
Combine $ x $ terms: $ -14x - 8x = -22x $
So the product is $ x^3 - 5x^2 - 22x + 56 $
(Note: Area arrays can be used by representing each binomial as a side of a rectangle and multiplying out the areas, but the distributive method is shown here for clarity.)
# Answer:
$ x^3 - 5x^2 - 22x + 56 $

#### (b)
# Explanation:
## Step1: Set the polynomial equal to zero
We have the polynomial $ (x - 2)(x + 4)(x - 7) = 0 $
## Step2: Use the zero - product property
By the zero - product property, if a product of factors is zero, then at least one factor is zero. So:
$ x - 2 = 0 $ or $ x + 4 = 0 $ or $ x - 7 = 0 $
Solving these equations:
For $ x - 2 = 0 $, $ x = 2 $
For $ x + 4 = 0 $, $ x = - 4 $
For $ x - 7 = 0 $, $ x = 7 $
This is known as the zero - product law (or zero - product property).
# Answer:
The three values of $ x $ are $ x = 2 $, $ x=-4 $, $ x = 7 $. This is known as the zero - product property (or zero - product law).

#### (c)
# Explanation:
## Step1: Choose a value, say $ x = 2 $
Substitute $ x = 2 $ into the polynomial $ (x - 2)(x + 4)(x - 7) $
## Step2: Calculate the value
$ (2 - 2)(2 + 4)(2 - 7)=0\times6\times(-5) = 0 $
So $ x = 2 $ is a zero of the cubic polynomial.
(We could also test $ x=-4 $: $ (-4 - 2)(-4 + 4)(-4 - 7)=(-6)\times0\times(-11)=0 $ or $ x = 7 $: $ (7 - 2)(7 + 4)(7 - 7)=5\times11\times0 = 0 $)
# Answer:
For example, when $ x = 2 $, $ (2 - 2)(2 + 4)(2 - 7)=0 $, so $ x = 2 $ is a zero of the cubic polynomial.

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QUESTION IMAGE

Question

Explanation:

Exercise #4

(a)

Step1: Substitute \( x = 4 \) into the binomials

Step2: Multiply the two results

Step1: Use the distributive property (FOIL method)

Step2: Combine like terms

Step1: Substitute \( x = 4 \) into the trinomial

Step2: Calculate the value

Answer:

(b)