What are the differences between the terms simplify , solve and factorize ?

[rosekidcute]

What are the differences between the terms simplify , solve and factorize ?

Are there any more terms like this i should be aware of when trying to learn some basic algebra ?

 

[ksdhart2]

The terms "simplify," "factorize," and "solve" do mean similar things, but the distinctions between them are important. For one thing, you can only "simplify" or "factorize" an expression, but you can only "solve" an equation. Thus, the distinction between "expression" and "equation" becomes important as well. The "rules" for what counts as an expression are very loose. A "real number" like 5, 12, or 123 is an expression. A single variable, like x, y, or z is also an expression. Expressions may also be more complex. x² - 5x - 6 and x³ + qx² + mzx + 67 are expressions. An equation, on the other hand, is simply any two expressions that are set equal to one another. y = mx + b is an equation, as is x² - 5x - 6 = 0. Note that the key difference between x² - 5x - 6, which is an expression, and x² - 5x - 6 = 0, which is an equation, is the presence of the equals symbol.

Now, if you were asked to simplify an expression, that's asking you to rewrite the expression in a different form that's easier to work with. For instance, 6/8 can be simplified by reducing it to lowest terms, making it 3/4. Factorizing is, in essence, a specific form of simplification, in which the expression is reduced to a form involving two (or more) factors. An example of that is factoring the quadratic x² - 5x - 6 to (x + 1)(x - 6). And, finally, solving an equation is when you find the value(s) for which the equation holds true. In some cases, the answer may be "all real numbers," in which case you may see that equation referred as an "identity."

Some examples of solving equations are x³ = 4x², which can be rewritten as x³ - 4x² = 0 and then factored down to x²(x - 4) = 0, produces the solutions x = 0 and x = 4. You may notice that, during the process of solving an equation, you often end up factorizing and/or simplifying the left-hand and/or right-hand expressions. Another example of solving an equation is sin(t) = 1, which gives the solution set $\displaystyle t = \pi + 2\pi \cdot n$ where n is any integer. This equation produces infinitely many solutions, yet they can be written compactly as one "family" of solutions. Finally, an example of an equation which has no solutions is $\displaystyle cos(m) = \pi$. This equation is never true (assuming real-valued m anyway), thus it has no solutions.

 

[rosekidcute]

Thanks a lot for the detailed explanations .

Few more things that has been confusing me are things like GCF and LCM of two numbers .

When we try to find the GCF or LCM of two numbers , expression or an equation , we are trying to simplify it right ?

 

[ksdhart2]

Thanks a lot for the detailed explanations .

Few more things that has been confusing me are things like GCF and LCM of two numbers .

When we try to find the GCF or LCM of two numbers , expression or an equation , we are trying to simplify it right ?

Not exactly, although finding the GCF or LCM of two (or more) numbers can be very useful when simplifying. GCF stands for Greatest Common Factor, and it is the largest number that is a factor of all of the original numbers. You may also see it written as GCD, with the D standing for Divisor. LCM stands for Least Common Multiple and is the smallest number that evenly divides all of the original numbers. You may also see this written as LCD, with the D standing for Denominator.

Some examples: The GCF of 128 and 900 can be found by breaking each of them down into their prime factors. 128 = 2⁷ and 900 = 2² * 3² * 5². The prime factors they share in common are 2², so their greatest common factor is 4. The GCF of 312, 900, and 1152 is 12, because 312 = 2³ * 3 * 13, 900 = 2² * 3² * 5², and 1152 = 2⁷ * 3². They share 2² and 3 as common factors, so their GCF is 2² * 3 = 12.

The LCM of 4 and 8 is 8, because 8 is the smallest number that evenly divides both 4 and 8. The LCM of 3 and 4 is 12, because 12 is the smallest number that evenly divides both 3 and 4. The LCM of two (or more) numbers will not always be equal to their product, but their product sets an upper bound. In other words, the LCM of some set of numbers will never be more than their product. The LCM of 4, 6, and 8 is only 24, even though their product is 192.

One way to find the LCM of some numbers is to again write out their prime factorization. Returning to the example of 312, 900, and 1152, we see that the LCM is equal to all of their prime factorizations multiplied together. 1152 has 2⁷ in its prime factorization. Out of the three numbers, this is the highest power of 2, so we know that will part of their LCM. Similarly, 3² is the highest shared power of 3, leaving 5² and 13 as non-shared powers. Thus, their LCM is 2⁷ * 3² * 5² * 13 = 374,400.

One place using the LCM is incredibly useful is for finding a common denominator. Suppose you wanted to add 5/7 + 6/11. To do so, you'd need to find their LCD of 77 and convert the fractions to share that denominator, and then you could add straight across: 55/77 + 42/77 = 97/77. The LCM can also be used with polynomial expressions. If you needed to find \(\displaystyle \dfrac{1}{x^2+3x+2} -
\dfrac{1}{x^2+5x+6}\), you'd find their LCD. You can do this by first factoring both, to get $\displaystyle \dfrac{1}{(x+1)(x+2)} - \dfrac{1}{(x+2)(x+3)}$. Thus, their LCD would be $\displaystyle (x+1)(x+2)(x+3)$, and the answer to the problem would be $\displaystyle \dfrac{1}{(x+1)(x+2)} - \dfrac{1}{(x+2)(x+3)} = \dfrac{2}{(x+1)(x+2)(x+3)}$.

 

[mmm4444bot]

"Solve" is also used as follows.

V = Pi * r^2 * h

Solve for h

In this context, the instruction to "solve for h" does not require finding a numerical value for h, but, rather, to re-express the relationship between V, r, h and Pi given in the equation so that symbol h is all by itself on one side of the equation.

(Some schools state this instruction as, "Make h the subject".)

Dividing each side of the given equation by Pi*r^2 accomplishes this.

h = V/(Pi * r^2)

The blue equation is a formula that gives you the volume of a right circular cylinder, if you know the radius and the height.

If, instead, you know the radius, but you want to determine the height required to hold a specific volume, then you would use the red equation (formula). :cool:

 

[Harry_the_cat]

Are their any more terms like this i should be aware of when trying to learn some basic algebra ?

Yes, make sure you know the difference between an expression, an equation and a function.

For example, in the realm of quadratics:

$\displaystyle ax^2 + bx + c$ is an expression (combo of numbers, variables and/or operation signs with NO equal sign). Expressions like this can be factorised.

$\displaystyle ax^2 + bx + c = 0$ is an equation (has an equal sign, doesn't have to be = 0). Equations like this can be solved.

$\displaystyle y = ax^2 + bx + c$ is a function, where, given an x-value, you can calculate a unique y-value. Functions like this can be graphed.

 

[mmm4444bot]

Are [there] any more terms like this i should be aware of …

"Expand"

This generally means to multiply algebraic factors, followed by simplifying (i.e., combining like-terms, reducing ratios).

Expand: (3x - 8)*(4x + 1)

12x^2 - 29x - 8


"Distribute"

When one factor is only a single term, and the other factor is composed of two or more terms inside grouping symbols, then the instruction "Distribute" is often used, instead of "Expand".

Distribute: a(b + c)

a*b + a*c

Both "expand" and "distribute" refer to multiplication.

Distributing and expanding are the opposite of factoring; that is, distribution and expansion each "undo" factoring. Likewise, factoring "undoes" expansion and distribution.

Factor: 12x^2 - 29x - 8

(3x - 8)*(4x + 1)


Factor: a*b + a*c

a(b + c) :cool:

 

[rosekidcute]

Thanks a lot for all the detailed replies .
I wish i had a book to practice Algebra , You know one with lot of Practice problems and its answers .

Anything like that is available out there ?

 

[mmm4444bot]

I wish i had a book to practice Algebra , You know one with lot of Practice problems and its answers .

Anything like that is available out there ?

Yes. Lots.

Start your search by googling the following keywords.

free online algebra books

free online algebra worksheets

free online algebra lessons exercises

Avoid any sites that ask for personal or payment information, and skip the ads. :cool:

 

[rosekidcute]

Thanks a lot for all the suggestions mmm4444bot