Math Struggles: Factorization & LCD

[Revolver]

I haven't taken a math class since high school and I'm 23 now. I jumped right into Precalc 1 for the summer and the first chapter kicked my ass. I completely forgot how to factor and do LCD with algebraic equations and my professor just breezes by it like its nothing.

Can anyone explain the general principles of factoring trinomial equations? Example x^2 + 5x 6 = 0, factor that. (Or is the term called FOIL?)

LCD is a problem too, I know the concept is multiply both denominators together to get the LCD to cancel out... but what if the equation is complex like (2+5x)/(x^2+4x+3) + (3-4x)/(5x+8) = 7x^2 + 7

another one was x/8 + 2x/4 = 25

Any help and or practice problems would be greatly appreciated, i need to catch up!

 

[HallsofIvy]

FOIL is a mnemonic for multiplying, the opposite of factoring. To multiply (x+a)(x+b) you have to multiply each part of the first term by each part of the second term. F is "first terms" x*x= x². O is "outside" x(b)= bx, I is "inside" a(x)= ax, L is "last" a(b)= ab, but the effect is simply that you multiplied the "x" in the first term by both x and b and the "a" in the first term by both x and b: (x+a)(x+b)= x²+ ax+ bx+ ab= x²+ (a+b)x+ ab. Now look carefully at the number multiplying x, a+ b, and the constant term ab. They are just the sum and product of the two numbers. To factor x² + 5x+ 6, work backwards. You know that ab must equal 6 so factor the 6 first: 6= 3(2). It also happens that 5= 3+2= a+b. x²+ 5x+ 6= (x+3)(x+2).
You do not "multiply both denominators together to get the LCD to cancel out". The point of the LCD "Least common denominator" is to not have to do that much work (unless absolutely necessary. To solve
$$\frac{2+ 5x}{x^2+4x+3}+\frac{3-4x}{5x+8}= 7x^2+ 7$$
factor the denominators: 3= 3(1) and 3+1= 4 so x²+ 4x+ 3= (x+3)(x+1). Since 5x+8 is not either of those, the LCD does happen to be the product: multiply each term of the equation by x²+ 4x+ 3 and 5x+ 8. Each of the denominators will cancel and you will have
[tex](2+5x)(5x+8)+ (3-4x)(x^2+ 4x+ 3)= (7x^2+ 7)(x^2+ 4x+ 3)(5x+8)[/itex]
The will be a fifth degree equation which might be impossible to solve exactly. I assume you just made that up, it's not an actual problem you were expected to solve.

For x/8+ 2x/4= 25, since 8= 2(4), just multiply each term by the LCD,8:
8(x/8)+ 8(2x)/4= 8(25), x+ 4x= 200, 5x= 200.

 

[Architeuthis Dux]

I understand the importance of a strong foundation in mathematics for success in many fields, including science. It is not uncommon for individuals to struggle with math concepts after a period of time away from the subject, and it is admirable that you are taking steps to catch up and improve your skills.

Factorization and finding the least common denominator (LCD) are fundamental concepts in algebra that are necessary for solving more complex equations. Let me provide some general principles and examples to help you better understand these concepts.

First, let's discuss factorization. The goal of factoring is to break down a polynomial expression into smaller parts, called factors. This is useful because it allows us to simplify expressions and solve equations. In the example you provided, x^2 + 5x + 6 = 0, we want to find the factors of 6 that add up to 5, which in this case are 2 and 3. So the expression can be factored as (x + 2)(x + 3) = 0. This means that either x + 2 = 0 or x + 3 = 0, giving us the solutions x = -2 and x = -3.

The term FOIL is often used as a mnemonic to help remember the steps for multiplying two binomials. It stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms. For example, to multiply (x + 2)(x + 3), we would first multiply the first terms (x * x), then the outer terms (x * 3), then the inner terms (2 * x), and finally the last terms (2 * 3). This results in x^2 + 5x + 6, which we can see is the same as our original expression.

Now let's discuss LCD. The LCD is the smallest number that is a multiple of all the denominators in a given equation. In the complex equation you provided, (2+5x)/(x^2+4x+3) + (3-4x)/(5x+8) = 7x^2 + 7, we first need to find the LCD of the two denominators, which in this case is (x+1)(5x+8). We then multiply both sides of the equation by this LCD to eliminate the fractions, resulting in a simpler equation to solve.

For