<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Concept. Go to the latest version.

# Special Products of Polynomials

## Work with multiple variables with special products of polynomials. Learn how to solve binomial squares, and the difference of squares with binomials.

Estimated8 minsto complete
%
Progress
Practice Special Products of Polynomials
Progress
Estimated8 minsto complete
%
Special Products of Polynomials

A flower is homozygous blue (RR) and another flower is homozygous white (rr). Use a Punnett square to show that a mixture of the two can produce a white flower.

### Guidance

There are two special cases of multiplying binomials. If you can learn to recognize them, you can multiply these binomials more quickly.

Here are the two special products that you should learn to recognize:

Special Case 1 (Binomial Squared): (x±y)2=x2±2xy+y2\begin{align*}(x\pm y)^2 = x^2 \pm 2xy + y^2\end{align*}

• Example: (x+5)2=x2+10x+25\begin{align*}(x+5)^2 = x^2 + 10x +25\end{align*}
• Example: (2x8)2=4x232x+64\begin{align*}(2x-8)^2 = 4x^2 -32x + 64\end{align*}

Special Case 2 (Difference of Perfect Squares): (x+y)(xy)=x2y2\begin{align*}(x+y)(x-y) = x^2 - y^2\end{align*}

• Example: (5x+10)(5x10)=25x2100\begin{align*}(5x+10)(5x-10) =25x^2 - 100\end{align*}
• Example: (2x4)(2x+4)=4x216\begin{align*}(2x-4)(2x+4)=4x^2-16\end{align*}

Keep in mind that you can always use the distributive property to do the multiplications if you don't notice that the problem is a special case.

#### Example A

Find the product: (x+11)2\begin{align*}(x+11)^2\end{align*}

Solution: This is an example of Special Case 1. You can use that pattern to quickly multiply.

(x+11)2=x2+2x11+112=x2+22x+121

You can verify that this is the correct answer by using the distributive property:

#### Example B

Find the product: (x7)2\begin{align*}(x-7)^2\end{align*}

Solution: This is another example of Special Case 1. You can use that pattern to quickly multiply.

(x7)2=x22x7+72=x214x+49

You can verify that this is the correct answer by using the distributive property:

#### Example C

Find the product: (x+9)(x9)\begin{align*}(x+9)(x-9)\end{align*}

Solution: This is an example of Special Case 2. You can use that pattern to quickly multiply.

(x+9)(x9)=x292=x281

You can verify that this is the correct answer by using the distributive property:

#### Concept Problem Revisited

Each flower will have one-half of the blue genes and one-half of the white genes. Therefore the equation formed will be:

0.5B+0.5W\begin{align*}0.5B + 0.5W\end{align*}

The offspring will have the genetic makeup (the mixture produced) using the equation:

(0.5B+0.5W)2\begin{align*}(0.5B + 0.5W)^2\end{align*}

Notice that this is an example of Special Case 1. You can expand the offspring genetic makeup equation to find out the percentage of offspring (or flowers) that will be blue, white, or light blue.

Therefore 25% of the offspring flowers will be blue, 50% will be light blue, and 25% will be white.

### Vocabulary

Distributive Property
The distributive property is a mathematical way of grouping terms. It states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. For example, in the expression: 3(x+5)\begin{align*}{\color{red}3}({\color{blue}x + 5})\end{align*}, the distributive property states that the product of a number (3)\begin{align*}({\color{red}3})\end{align*} and a sum (x+5)\begin{align*}({\color{blue}x + 5})\end{align*} is equal to the sum of the individual products of the number (3)\begin{align*}({\color{red}3})\end{align*} and the addends (x\begin{align*}({\color{blue}x}\end{align*} and 5)\begin{align*}{\color{blue}5})\end{align*}.

### Guided Practice

1. Expand the following binomial: (x+4)2\begin{align*}(x+4)^2\end{align*}.

2. Expand the following binomial: (5x3)2\begin{align*}(5x -3)^2\end{align*}.

3. Determine whether or not each of the following is a difference of two perfect squares:

a) a216\begin{align*}a^2-16\end{align*}

b) 9b249\begin{align*}9b^2-49\end{align*}

c) c260\begin{align*}c^2-60\end{align*}

1. (x+4)2=x2+4x+4x+16=x2+8x+16\begin{align*}(x+4)^2=x^2+4x+4x+16=x^2+8x+16\end{align*}.

2. (5x3)2=25x215x15x+9=25x230x+9\begin{align*}(5x-3)^2=25x^2-15x-15x+9=25x^2-30x+9\end{align*}

3. a)Yes, a216=(a+4)(a4)\begin{align*}a^2 - 16 = (a+4)(a-4)\end{align*}

b) Yes, 9b249=(3b+7)(3b7)\begin{align*}9b^2 - 49 = (3b +7)(3b - 7)\end{align*}
c) No, 60\begin{align*}60\end{align*} is not a perfect square.

### Practice

Expand the following binomials:

1. (t+12)2\begin{align*}(t+12)^2\end{align*}
2. (w+15)2\begin{align*}(w+15)^2\end{align*}
3. (2e+7)2\begin{align*}(2e+7)^2\end{align*}
4. (3z+2)2\begin{align*}(3z+2)^2\end{align*}
5. (7m+6)2\begin{align*}(7m+6)^2\end{align*}
6. (g6)2\begin{align*}(g-6)^2\end{align*}
7. (d15)2\begin{align*}(d-15)^2\end{align*}
8. (4x3)2\begin{align*}(4x-3)^2\end{align*}
9. (2p5)2\begin{align*}(2p-5)^2\end{align*}
10. (6t7)2\begin{align*}(6t-7)^2\end{align*}

Find the product of the following binomials:

1. (x+13)(x13)\begin{align*}(x+13)(x-13)\end{align*}
2. (x+6)(x6)\begin{align*}(x+6)(x-6)\end{align*}
3. (2x+5)(2x5)\begin{align*}(2x+5)(2x-5)\end{align*}
4. (3x+4)(3x4)\begin{align*}(3x+4)(3x-4)\end{align*}
5. (6x+7)(6x7)\begin{align*}(6x+7)(6x-7)\end{align*}

### Vocabulary Language: English

Binomial

Binomial

A binomial is an expression with two terms. The prefix 'bi' means 'two'.
distributive property

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
Square of a Binomial

Square of a Binomial

The product of a squared binomial is always a trinomial.