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# Multiplying Special Cases

Sep 15, 2022

## Key Concepts

• Determine the square of a binomial
• Find the product of a sum and a difference
• Apply the square of a binomial

## Determine the Square of a Binomial

### Concept

Why is (a + b)2 considered a special case when multiplying polynomials?

Use the distributive property.

Use a visual model.

The square of a binomial follows the pattern: (a + b)2 = a2 + 2ab + b2

Example 1

What is the product (5x – 3)2

Use the pattern learned in the previous slide to find the square of a difference.

(5x – 3)2 = (5x + (-3))2 ——————————Rewrite the difference as sum.

= (5x)2+ 2(5x)(-3) + (-3)2————Substitute 5x and -3 into a2+ 2ab + b2

= 25x2– 30x + 9 ————————-Simplify

You can write the product (5x – 3) 2as 25x2– 30x + 9.

Example 2

How can you use the square of a binomial to find the product 292

Rewrite the product as a difference of two values whose squares you know, such as (30 – 1) 2. Then use the pattern for the square of a binomial to find its square.

So, 292 = 841. In general, you can use the square of a binomial to find the square of a large number by rewriting the number as the sum or difference of two numbers with known squares.

## Find the Product of a Sum and a Difference

### Concept

What is the product (a + b)(a – b)?

Use the distributive property to find the product.

The product of two binomials in the form: (a + b)(a – b) is a2 – b2

The product of the sum and the difference of the same two values results in the difference of two squares.

Example 3

What is the product (5x + 7) (5x – 7)?

Use the pattern learned in the previous slide.

(5x + 7) (5x – 7) = (5x)2 – (7)2——— Substitute 5x and 7 into a2– b2.

= 25x2– 49 ———— Simplify.

The product of (5x + 7) (5x – 7) is 25x2– 49. It is the difference of two squares, (5x)2 – (7)2.

Example 4

How can you use the difference of two squares to find the product of 43 and 37?

Rewrite the product as the sum and the difference of the same two numbers, a and b.

You can use the difference of two squares to mentally find the product of large numbers when the numbers are the same distance from a known square.

### Apply the Square of a Binomial

Example 5

A graphic designer is developing images for icons. The square pixelated image is placed inside a border that is 2 pixels wide on all sides. If the area of the border of the image is 176 square pixels, what is the area of the image?

Solution:

Let x represent the length and the width of the image.

Formulate:

The area of the image and the border is represented by the expression (x + 4)2.

Compute:

The image will be 20 pixels by 20 pixels. The area of the image is 20*20, or 400 square pixels.

### Questions

Question 1

Find each product.

1. (3x – 4)2

Solution:

Using the pattern (a + b) 2 = a2 + 2ab + b2

(3x + (-4)) 2 = (3x) 2 + 2 × 3x × (-4) + (-4) 2 = 9x2 – 24x + 16

2. 712

Solution:

(70 + 1) 2 = (70) 2 + 2 × 70 ×1 + (1) 2 = 4900 + 140 + 1 = 5041

Question 2

Find each product.

1. (2x – 4)(2x + 4)

Solution:

Using the pattern (a + b)(a – b) = a2 – b2

(2x – 4)(2x + 4) = (2x) 2 – 42 = 4x2 – 16

2. 56*44

Solution:

(50 + 6)(50 – 6) = (50) 2 – 62 = 2500 – 36 = 2464

Question 3

In example 5, what is the area of the square image if the area of the border is 704 square pixels and the border is 4 pixels wide?

Solution:

Let x represent the length and the width of the image.

Since the border is 4 pixels wide,

The area of the image and the border is represented by the expression (x + 8)2.

Area of border = total area – area of image

= (x + 8) 2 – x2

= x2 + 2 × 8 × x + 82 – x2

= 16x + 64

Since the area of border is 704 square pixels,

16x + 64 = 704

16x = 640

x = 40 pixels

Area of the image = x2 = 402 = 1600 square pixels

## Exercise

1. Find the product of the following equations:
• (x+9) (x-9)
• (x-7) (x-7)
• (2x – 1) 2
• (x – 7) 2
• (2x + 5) 2
2. Kennedy multiplies (x – 3) (x + 3) and gets an answer of x2 – 6x – 9. Describe and correct the Kennedy’s error.
3. Use the square of a binomial to find the product. 542
4. Explain why the product of two binomials in the form (a + b) (a – b) is a binomial instead of a trinomial.
5. Find the product. (3a – 4b) (3a + 4b)
6. Find the product. (x2 – 2y) (x2 + 2y)

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