7.3: 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.
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Khan Academy Special Products of Binomials
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): \begin{align*}(x\pm y)^2 = x^2 \pm 2xy + y^2\end{align*}
- Example: \begin{align*}(x+5)^2 = x^2 + 10x +25\end{align*}
- Example: \begin{align*}(2x-8)^2 = 4x^2 -32x + 64\end{align*}
Special Case 2 (Difference of Perfect Squares): \begin{align*}(x+y)(x-y) = x^2 - y^2\end{align*}
- Example: \begin{align*}(5x+10)(5x-10) =25x^2 - 100\end{align*}
- Example: \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: \begin{align*}(x+11)^2\end{align*}
Solution: These is an example of Special Case 1. You can use that pattern to quickly multiply.
\begin{align*}(x+11)^2&=x^2+2\cdot x\cdot 11 +11^2\\ &=x^2+22x+121\end{align*}
You can verify that this is the correct answer by using the distributive property:
Example B
Find the product: \begin{align*}(x-7)^2\end{align*}
Solution: These is another example of Special Case 1. You can use that pattern to quickly multiply.
\begin{align*}(x-7)^2&=x^2-2\cdot x\cdot 7 +7^2\\ &=x^2-14x+49\end{align*}
You can verify that this is the correct answer by using the distributive property:
Example C
Find the product: \begin{align*}(x+9)(x-9)\end{align*}
Solution: These is an example of Special Case 2. You can use that pattern to quickly multiply.
\begin{align*}(x+9)(x-9)&=x^2-9^2\\ &=x^2-81\end{align*}
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:
\begin{align*}0.5B + 0.5W\end{align*}
The offspring will have the genetic makeup (the mixture produced) using the equation:
\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: \begin{align*}{\color{red}3}({\color{blue}x + 5})\end{align*}, the distributive property states that the product of a number \begin{align*}({\color{red}3})\end{align*} and a sum \begin{align*}({\color{blue}x + 5})\end{align*} is equal to the sum of the individual products of the number \begin{align*}({\color{red}3})\end{align*} and the addends \begin{align*}({\color{blue}x}\end{align*} and \begin{align*}{\color{blue}5})\end{align*}.
Guided Practice
1. Expand the following binomial: \begin{align*}(x+4)^2\end{align*}.
2. Expand the following binomial: \begin{align*}(5x -3)^2\end{align*}.
3. Determine whether or not each of the following is a difference of two perfect squares:
a) \begin{align*}a^2-16\end{align*}
b) \begin{align*}9b^2-49\end{align*}
c) \begin{align*}c^2-60\end{align*}
Answers:
1. \begin{align*}(x+4)^2=x^2+4x+4x+16=x^2+8x+16\end{align*}.
2. \begin{align*}(5x-3)^2=25x^2-15x-15x+9=25x^2-30x+9\end{align*}
3. a)Yes, \begin{align*}a^2 - 16 = (a+4)(a-4)\end{align*}
- b) Yes, \begin{align*}9b^2 - 49 = (3b +7)(3b - 7)\end{align*}
- c) No, \begin{align*}60\end{align*} is not a perfect square.
Practice
Expand the following binomials:
- \begin{align*}(t+12)^2\end{align*}
- \begin{align*}(w+15)^2\end{align*}
- \begin{align*}(2e+7)^2\end{align*}
- \begin{align*}(3z+2)^2\end{align*}
- \begin{align*}(7m+6)^2\end{align*}
- \begin{align*}(g-6)^2\end{align*}
- \begin{align*}(d-15)^2\end{align*}
- \begin{align*}(4x-3)^2\end{align*}
- \begin{align*}(2p-5)^2\end{align*}
- \begin{align*}(6t-7)^2\end{align*}
Find the product of the following binomials:
- \begin{align*}(x+13)(x-13)\end{align*}
- \begin{align*}(x+6)(x-6)\end{align*}
- \begin{align*}(2x+5)(2x-5)\end{align*}
- \begin{align*}(3x+4)(3x-4)\end{align*}
- \begin{align*}(6x+7)(6x-7)\end{align*}
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Binomial
A binomial is an expression with two terms. The prefix 'bi' means 'two'.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, .Square of a Binomial
The product of a squared binomial is always a trinomial.Image Attributions
Here you will learn about special cases of binomial multiplication.