Question 1: Factor the expression as a product of binomials.

*x*2 + 14*x* + 45

Question 2: Factor the expression as a product of binomials.

*x*2 − 21*x* + 20

Question 3: Factor the expression as a product of binomials.

*x*2 − 14*x* + 49

Question 4: Factor completely.

64*x*2 − *x*

Question 5: Factor completely.

3*x*5 + 12*x*3

Question 6: Factor completely.

2*x*2 − 2*x* − 112

Question 7: Revenue *R* from the sale of *x* units of a product is found by multiplying the price by the number of items sold.

(a) Factor the right side of 

*R* = 800*x* − *x*2.

*R* = 

(b) What is the expression for the price of the item?

Question 8: The consumer expenditure, in dollars, for a commodity is the product of its market price, *p*, and the number of units demanded, *x*. Suppose that for a certain commodity, the consumer expenditure is given by 

13,000*p* − 100*p*2.

(a) Factor this in order to find an expression for the number of units demanded. (Simplify your answer completely.)
*x* = 
 
 units

(b) Use (a) to find the number of units demanded when the market price is $42.
*x* =  units


Sure, let's solve each of these questions step by step.

### Question 1:
Factor the expression as a product of binomials: $ x^2 + 14x + 45 $

To factor the quadratic expression $ x^2 + 14x + 45 $, we look for two numbers that multiply to 45 and add to 14. These numbers are 5 and 9.

Thus,
\[ x^2 + 14x + 45 = (x + 5)(x + 9) \]

### Question 2:
Factor the expression as a product of binomials: $ x^2 - 21x + 20 $

To factor the quadratic expression $ x^2 - 21x + 20 $, we look for two numbers that multiply to 20 and add to -21. These numbers are -1 and -20.

Thus,
\[ x^2 - 21x + 20 = (x - 1)(x - 20) \]

### Question 3:
Factor the expression as a product of binomials: $ x^2 - 14x + 49 $

To factor the quadratic expression $ x^2 - 14x + 49 $, we look for two numbers that multiply to 49 and add to -14. These numbers are -7 and -7.

Thus,
\[ x^2 - 14x + 49 = (x - 7)(x - 7) = (x - 7)^2 \]

### Question 4:
Factor completely: $ 64x^2 - x $

First, factor out the greatest common factor (GCF), which is $ x $.

\[ 64x^2 - x = x(64x - 1) \]

### Question 5:
Factor completely: $ 3x^5 + 12x^3 $

First, factor out the greatest common factor (GCF), which is $ 3x^3 $.

\[ 3x^5 + 12x^3 = 3x^3(x^2 + 4) \]

### Question 6:
Factor completely: $ 2x^2 - 2x - 112 $

First, factor out the greatest common factor (GCF), which is $ 2 $.

\[ 2(x^2 - x - 56) \]

Now, we need to factor the quadratic $ x^2 - x - 56 $. We look for two numbers that multiply to -56 and add to -1. These numbers are 7 and -8.

Thus,
\[ x^2 - x - 56 = (x - 8)(x + 7) \]

So, 
\[ 2x^2 - 2x - 112 = 2(x - 8)(x + 7) \]

### Question 7:
Revenue $ R $ from the sale of $ x $ units of a product is found by multiplying the price by the number of items sold.

#### (a) Factor the right side of $ R = 800x - x^2 $.

First, factor out $ x $.

\[ R = x(800 - x) \]

#### (b) What is the expression for the price of the item?

The expression for the price of the item is $ 800 - x $.

### Question 8:
The consumer expenditure, in dollars, for a commodity is the product of its market price, $ p $, and the number of units demanded, $ x $.

#### (a) Factor this in order to find an expression for the number of units demanded:
\[ 13,000p - 100p^2 \]

First, factor out the greatest common factor (GCF), which is $ 100p $.

\[ 13,000p - 100p^2 = 100p(130 - p) \]

Thus, the number of units demanded $ x $ is given by:
\[ x = 100(130 - p) \]

#### (b) Use (a) to find the number of units demanded when the market price is $42.
\[ x = 100(130 - 42) \]
\[ x = 100(88) \]
\[ x = 8800 \]

The number of units demanded when the market price is $42 is 8800 units.

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### Related Questions:
1. How do you verify the factored form of a quadratic expression?
2. What are the special cases in factoring quadratic expressions?
3. How can we use factoring to solve quadratic equations?
4. What are the common mistakes to avoid while factoring polynomials?
5. How is factoring used in solving real-world problems involving quadratic expressions?

### Tip:
When factoring quadratic expressions, always look for the greatest common factor first before applying other factoring techniques.

Question 1: Factor the expression as a product of binomials.

x^2 + 14x + 45

Question 2: Factor the expression as a product of binomials.

x^2 − 21x + 20

Question 3: Factor the expression as a product of binomials.

x^2 − 14x + 49

Question 4: Factor completely.

64x^2 − x

Question 5: Factor completely.

3x^5 + 12x^3

Question 6: Factor completely.

2x^2 − 2x − 112

Question 7: Revenue R from the sale of x units of a product is found by multiplying the price by the number of items sold.

(a) Factor the right side of R = 800x − x^2.

(b) What is the expression for the price of the item?

Question 8: The consumer expenditure, in dollars, for a commodity is the product of its market price, p, and the number of units demanded, x. Suppose that for a certain commodity, the consumer expenditure is given by 13,000p − 100p^2.

(a) Factor this in order to find an expression for the number of units demanded.

(b) Use (a) to find the number of units demanded when the market price is $42.

Explore how to factor quadratic expressions and polynomials, including real-world applications involving revenue and consumer expenditure. This problem set covers algebraic concepts such as binomial factoring and the quadratic formula. Suitable for Grades 8-10.

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