Use algebraic manipulation to prove the distributivity property

In summary, the distributive rule is used to prove that x+yz=(x+y)(x+z). By expanding the right side using the distributive rule, we can see that it simplifies to x+yz, proving the equation.
  • #1
shamieh
539
0
Use algebraic manipulation to prove that x+yz=(x+y)(x+z) Note that this is
the distributive rule,

So I have:

(x + y)(x + z) = xx + xz + xy + zy
= x + xz + xy + zy
= x(1 + z + y) zy
=x * 1 + (z + y)(zy)
= x + ((zzy + yzz)
= x +y + z?

but i need x + yz... But I don't understand what is going on in the right side
 
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  • #2
I don't understand what you are doing, either.

(x + y)(x + z) = (x + y)x + (x + y)z

= xx + yx + xz + yz = xx + xy + xz + yz (since xy = yx)

= x + xy + xz + yz (since xx = x)

= x(1 + y + z) + yz =...?
 

FAQ: Use algebraic manipulation to prove the distributivity property

What is the distributivity property?

The distributivity property is a fundamental property of algebra that states that the product of a number and the sum or difference of two or more numbers is equal to the sum or difference of the products of that number and each of the individual numbers. In other words, a(b + c) = ab + ac and a(b - c) = ab - ac.

Why is the distributivity property important?

The distributivity property is important because it allows us to simplify and manipulate algebraic expressions, making them easier to solve. It is also used in more advanced mathematical concepts, such as factoring and expanding polynomials, and in various real-world applications such as finance and physics.

How do you prove the distributivity property?

The distributivity property can be proved using algebraic manipulation. This involves breaking down the expression into smaller parts and showing that they are equivalent. For example, to prove a(b + c) = ab + ac, we can expand the left side of the equation to get ab + ac, which is equal to the right side of the equation. This shows that the distributivity property holds true.

Can the distributivity property be applied to any set of numbers?

Yes, the distributivity property can be applied to any set of numbers, including integers, fractions, decimals, and even complex numbers. As long as the numbers follow the basic rules of arithmetic, the distributivity property will hold true.

Are there any exceptions to the distributivity property?

No, the distributivity property holds true for all numbers and variables. However, it is important to note that it only applies to multiplication and addition or subtraction, not to other operations such as division or exponentiation. Additionally, it cannot be applied to non-numeric values, such as matrices or functions.

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