Consider the following binomials.
A=(x^2+5x) B=(6x+30)
Part 1: Factor each binomial by finding the GCF. Then, add the two factored binomials to make a single expression.
Part 2: Now add the original forms of binomial A and B together to make a trinomial. Factor the trinomial.
Part 3: Do you see something you could do to your answer in Part A to get your answer to Part B? Explain.
I know how to factor the binomials by finding the GCF. But I'm confused in this case for Parts A and B. For B, I think the gcf can be 3 and for A, the GCF can be 5.
1. The GCF is x + 5
A = (x+5)x and b = 6(x+5)
2. A + B = x^2 + 11 x + 30 = (x+6)(x+5)
3. Looking at the answer in (1), it can also be written (x+5)(x+6), with the GCF as one of the factors.
For Part 1, let's start with factoring each binomial by finding the GCF:
A = x^2 + 5x
To find the GCF, we look for common factors in both terms, which are x:
x(x + 5)
B = 6x + 30
The GCF here is 6:
6(x + 5)
Now, we can add the two factored binomials together to make a single expression:
x(x + 5) + 6(x + 5)
Combining the terms inside the parentheses:
(x + 6)(x + 5)
For Part 2, we're asked to add the original forms of binomial A and B together to make a trinomial. Then, we need to factor the trinomial:
A = (x^2 + 5x)
B = (6x + 30)
Adding A and B together:
(x^2 + 5x) + (6x + 30)
Combining like terms:
x^2 + 5x + 6x + 30
Simplifying:
x^2 + 11x + 30
Now, let's factor the trinomial:
We'll look for two numbers that multiply to give us 30 and add up to 11.
The numbers are 5 and 6:
(x + 5)(x + 6)
For Part 3, to go from the answer in Part A (x + 6)(x + 5) to the answer in Part B (x + 5)(x + 6), we noticed that the order of the factors got switched. In algebra, the order of the factors doesn't affect the result. So, we can multiply (x + 6)(x + 5) in Part A and still end up with (x + 5)(x + 6), which is the same as the factored form of the trinomial in Part B.
To factor the binomials A and B by finding the greatest common factor (GCF), let's break down the process step by step:
Part 1:
First, let's find the GCF for binomial A = (x^2 + 5x).
The GCF of the terms "x^2" and "5x" is "x" since it is the largest factor that divides evenly into both terms.
Now we can factor out this GCF of "x" from each term:
A = x(x + 5)
Next, let's find the GCF for binomial B = (6x + 30).
Notice that both terms in B are divisible by 6. Also, both terms can be divided by 6x, resulting in a GCF of 6x.
Now we can factor out this GCF of 6x from each term:
B = 6x(1 + 5)
Simplifying the second term within the parentheses:
B = 6x(6)
To combine the two factored binomials A and B into a single expression, we add them together:
A + B = x(x + 5) + 6x(6)
Part 2:
Now, let's add the original forms of binomial A and B together to make a trinomial:
A + B = (x^2 + 5x) + (6x + 30)
Combining like terms, we get:
A + B = x^2 + 5x + 6x + 30
Simplifying further:
A + B = x^2 + 11x + 30
To factor this trinomial, we are looking for two binomials whose product is equal to the given trinomial.
Part 3:
In Part A, we factored both binomials A and B individually. In Part B, we added the original forms of A and B together and then factored the resulting trinomial.
If you compare the factored form of binomial A (x(x + 5)) from Part A and the factored form of the trinomial A + B (x^2 + 11x + 30) from Part B, you will notice that they are the same.
This means that we could have directly factored the trinomial A + B by identifying the GCF of the original binomials A and B.
Therefore, to go from Part A to Part B, you can factor the trinomial immediately without adding the individual terms of binomials A and B together. This saves a step in the process.