15x^3-9x^2-6x
Can someone please show me how to factor completely.. trying to learn how to do factoring
Usually factoring by grouping is useful when patterns jump out at you:
x^3 + 3x^2 - 6x - 18
you can see x(x^2-6) + 3(x^2-6)
In your problem, that's not likely, so grouping is not a useful method. Instead, factor out the 3x to get
3x(5x^2 - 3x - 2)
Now, if you want to use grouping, you have to play around a bit:
5x^2-3x-2 = 5x^2-5x + 2x-2
= 5x(x-1) + 2(x-1)
= (5x+2)(x-1)
Again, picking up on how to group is a skill learned through experience. Just buckle down and do a few hundred exercises, and you too will be proficient!
Of course! Factoring is the process of expressing a polynomial as a product of its factors. To factor the polynomial 15x^3 - 9x^2 - 6x completely, we'll start by looking for common factors among the terms.
Step 1: Identify the Greatest Common Factor (GCF)
In this case, the GCF is 3x, as it's the largest factor that divides evenly into each term. We can factor it out by dividing each term by 3x:
3x(5x^2 - 3x - 2)
Step 2: Factor the Trinomial
Now, we'll focus on factoring the trinomial, (5x^2 - 3x - 2). We're looking for two binomials that, when multiplied together, result in the given trinomial. To do this, we'll use the method known as "ac-method" or "product-sum method".
The ac-method involves looking at the coefficient of the x^2 term (a), the constant term (c), and finding two numbers that multiply to give ac (5 * -2 = -10) and add up to b (-3). In this case, the numbers are -5 and 2, since (-5) * 2 = -10 and (-5) + 2 = -3.
Now, we'll rewrite the middle term (-3x) using these two numbers:
3x(5x^2 - 5x + 2x - 2)
Step 3: Group and Factor
Next, we group the terms together by pairs and factor out the common factors:
3x( (5x^2 - 5x) + (2x - 2) )
3x(5x(x - 1) + 2(x - 1))
Step 4: Factor out the common binomial
Now, we can see that (x - 1) is a common binomial factor in both terms:
3x( (5x + 2)(x - 1) )
So, the completely factored form of 15x^3 - 9x^2 - 6x is 3x(5x + 2)(x - 1).