(r^2+7r+10)/3 * (3r-30)/(r^2-5r-50)
how can i Factorize each expression to end up with this new equation
(r+5)(r+2)/3 * 3(r-10)/((r-10)(r+5)
can you please explain..
I know that they cancel but what i want to know its how did this expression
(r^2+7r+10)/3 * (3r-30)/(r^2-5r-50)
became this (r+5)(r+2)/3 * 3(r-10)/((r-10)(r+5)
like the first expression had a 7r where is that in the second.
Let's factor r^2 + 7r + 10.
When we factor a trinomial, we get the
product of 2 binomials.
1. First, we replace r^2 with r*r:
(r + )(r + ),
2. We eliminate 7r, and 10 by finding 2 numbers whose product equals 10 and
their sum equals 7:
10 = 1*10 = 2*5.
We select 2, and 5 because their sum equals 7 and their product equals 10.
The factored trinominal equals:
(r + 2) (r + 5)
This method of factoring is convient only when the coefficient of X^2 = 1.
CHECK: Multiply each term in the 1st
parenthesis by the quantity in the 2nd parenthesis:
r(r + 5) + 2(r + 5),
Do the Multiplication:
r^2 + 5r + 2r + 10,
Combine like-terms and get:
r^2 + 7r + 10.
You try factoring the 2nd trinomial.
HINT: -50 = 1(-50) = 2(-25) = 5(-10).
To factorize each expression, let's start with the first expression: (r^2 + 7r + 10)/3.
1. We need to find two numbers that sum up to 7 and multiply to give 10 (the coefficients of 'r^2' and the constant term). These numbers are 5 and 2.
So, we can rewrite the numerator as: (r + 5)(r + 2)/3.
Now let's move on to the second expression: (3r - 30)/(r^2 - 5r - 50).
2. Factorizing the numerator:
We can factor out a common factor of 3 from 3r and -30:
3(r - 10)/(r^2 - 5r - 50).
3. Factorizing the denominator:
We need to find two numbers that sum up to -5 and multiply to give -50 (the coefficients of 'r^2' and the constant term). These numbers are -10 and 5.
So, we can rewrite the denominator as: (r - 10)(r + 5).
Combining both the numerator and denominator, we have:
[(r + 5)(r + 2)/3] * [3(r - 10)/((r - 10)(r + 5))].
Note: Here, we canceled out the common factors of (r - 10) and (r + 5). Be careful when canceling out factors, as there may be restrictions on the values of 'r' that make the equation undefined (such as when the denominator is zero).
In this case, the factor (r - 10) cancels out, and we're left with:
(r + 5)(r + 2)/3 * 3/1.
Simplifying further, we get:
(r + 5)(r + 2) *(3/3).
Finally, canceling out the common factor of 3/3, we end up with:
(r + 5)(r + 2)/1, which can be simplified to:
(r + 5)(r + 2).