Factor the following polynomial6x3+9x2−15x6x^3+9x^2-15x

3 x ( x − 1 ) ( 2 x + 5 ) 3x\left(x-1\right)\left(2x+5\right) 3 x ( x − 1 ) ( 2 x + 5 )

Factor the following polynomial 6 x 3 + 9 x 2 − 15 x 6x^3+9x^2-15x

For this problem, we've been asked to factor this polynomial. Now the first thing I noticed about this polynomial is that all of the coefficients are multiples of three, and that all of the terms in this polynomial share a variable of at least one x. So that tells me that this polynomial has a greatest common factor of three x, and I can factor that out first before applying any of my other factoring techniques. So factoring out three x, we're left with two x squared plus three x minus five. And now I can apply my factoring techniques to this trinomial. For this one, we're going to use our AC method, also known as our grouping method. So first, let's identify our a and c value, as well as our b value. So we want to find two numbers that multiply to our a times c value of negative 10 and also add to our b value of three. So let's look at some factors of negative 10. Well, there's two and negative five, but those sum to negative three. So that's not what we're looking for. But if we switch the signs and do negative two and positive five, then we do get positive three. So this is the factor pair that we want. And now I can split apart this middle term of the trinomial to turn this three term polynomial into a four term polynomial. Doing that, we have our three x times two x squared minus two x plus five x minus five. And now I can apply my factoring by grouping technique. So let's first try grouping our first two terms and last two terms and see if they produce a GCF. For my first two terms, they do have a GCF of two x. And for my last two terms, we also have a GCF here of five. So now I can factor those out as well. So now we have a three x all times two x times x minus one plus five times also x minus one. And now that we have a common binomial, we can also factor it out. So now we have three x times x minus one times two x plus five. And this here is our final answer.

Factor the following polynomial 6 x 3 + 9 x 2 − 15 x 6x^3+9x^2-15x

3 x ( x − 1 ) ( 2 x + 5 ) 3x\left(x-1\right)\left(2x+5\right) 3 x ( x − 1 ) ( 2 x + 5 )

3 x ( x + 1 ) ( 2 x − 5 ) 3x\left(x+1\right)\left(2x-5\right) 3 x ( x + 1 ) ( 2 x − 5 )

3 x ( 2 x − 5 ) ( x − 1 ) 3x\left(2x-5\right)\left(x-1\right) 3 x ( 2 x − 5 ) ( x − 1 )

( x − 1 ) ( 2 x + 5 ) \left(x-1\right)\left(2x+5\right) ( x − 1 ) ( 2 x + 5 )

7. Factoring

Factoring Trinomials of the Form ax² + bx + c

Intermediate Algebra

7. Factoring

Factoring Trinomials of the Form ax² + bx + c

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Multiple Choice

Factor the following polynomial
$6 x^{3} + 9 x^{2} - 15 x$

$3x$\left$(x+1$\right$)$\left$(2x-5$\right$)$

$3x$\left$(x-1$\right$)$\left$(2x+5$\right$)$

$3x$\left$(2x-5$\right$)$\left$(x-1$\right$)$

$$\left$(x-1$\right$)$\left$(2x+5$\right$)$

Verified step by step guidance

Identify the greatest common factor (GCF) of all the terms in the polynomial \$6x^3 + 9x^2 - 15x$. Look at the coefficients (6, 9, and 15) and the variable parts ($x^3$, $x^2$, and $x$) to find the GCF.

Factor out the GCF from each term of the polynomial. This means dividing each term by the GCF and writing the polynomial as the product of the GCF and the resulting simplified polynomial.

Focus on factoring the remaining polynomial inside the parentheses after factoring out the GCF. The polynomial inside will be a quadratic expression in terms of $x$.

Use factoring techniques for quadratics, such as finding two numbers that multiply to the product of the quadratic coefficient and the constant term, and add to the middle term's coefficient. Then rewrite the quadratic as a product of two binomials.

Write the fully factored form of the original polynomial as the product of the GCF and the two binomials found in the previous step.

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