Factor completely.15a2+25a−4015a^2+25a-40

5 ( 3 a + 8 ) ( a − 1 ) 5\left(3a+8\right)\left(a-1\right)

Factor completely. 15 a 2 + 25 a − 40 15a^2+25a-40

Here we've been asked to factor this trinomial completely, and the first thing that I notice is that these terms are quite large and they all seem to be multiples of five, which tells me that this trinomial has a greatest common factor of five, and we can start off by factoring that out. Factoring out five, we have 15 divided by a squared, so we'd be left with three a squared. 25 divided by five is five, so we have five a. And 40 divided by five is eight, so we have minus eight. And now I can apply my trial and error method to this much smaller trinomial. So our first step here is to write out the factors of three, which are three and one. And next, we'll write out the factors of negative eight, which are negative eight and one, positive eight and negative one, as well as negative two, positive four, and positive two and negative four. And now we can come up with some possible binomial factor pairs. So first, let's try pairing up three a and one a with negative eight and positive one. Now remember, here we only need to focus on the middle two steps of our foil technique because that's what produces the middle value in our trinomial. So doing this we have 3a times one which is 3a and next negative eight times 1a which is negative 8a and this sums to negative 5a which is the right number, but not the correct value as we need positive five a. So to fix this, we're simply going to change the signs of the last terms of each binomial. Doing that, we have three a plus eight times one a minus one. And now applying our foil technique, we have three a times negative one, so we have negative three a, and eight times a positive one a, so we have plus eight a, which sums to positive five a, which is the value that we're looking for. So that tells me that I can factor this trinomial as a three a plus eight times one a minus one.

Factor completely. 15 a 2 + 25 a − 40 15a^2+25a-40

5 ( 3 a + 8 ) ( a − 1 ) 5\left(3a+8\right)\left(a-1\right)

( 3 a + 8 ) ( a − 1 ) \left(3a+8\right)\left(a-1\right)

5 ( 3 a − 8 ) ( a − 1 ) 5\left(3a-8\right)\left(a-1\right)

( 3 a + 8 ) ( a + 1 ) \left(3a+8\right)\left(a+1\right)

7. Factoring

Factoring Trinomials of the Form ax² + bx + c

Beginning & Intermediate Algebra

7. Factoring

Factoring Trinomials of the Form ax² + bx + c

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Multiple Choice

Factor completely.
$15 a^{2} + 25 a - 40$

$(3 a + 8) (a - 1)$

$5 (3 a - 8) (a - 1)$

$(3 a + 8) (a + 1)$

$5 (3 a + 8) (a - 1)$

Verified step by step guidance

First, identify the greatest common factor (GCF) of all the terms in the expression \$15a^2 + 25a - 40$. The GCF is the largest number that divides evenly into 15, 25, and 40.

Factor out the GCF from the entire expression. This means rewriting the expression as the GCF multiplied by a trinomial with smaller coefficients.

Next, focus on factoring the trinomial inside the parentheses. Look for two numbers that multiply to the product of the coefficient of $a^2$ term and the constant term (after factoring out the GCF) and add up to the coefficient of the $a$ term.

Use these two numbers to split the middle term and then factor by grouping. Group the terms in pairs and factor out the common binomial factor from each group.

Write the final factored form as the product of the GCF and the two binomials obtained from factoring by grouping.

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