Oh God... Don't laugh, but how do I factor polynomials with a coefficient (sp?) in front of the first monomial

Question

Something like
3x^2+15x+5

Ok, that one probably wouldn't factor out evenly... But you guys get it, right?

Answer 1

Of course, if a quadratic simply MUST be factored, there's always the brute force method of factoring a polynomial, which is to factor ax² + bx + c as a(x-r₁)(x-r₂), where r₁ and r₂ are the two roots of the polynomial ax² + bx + c. So in this case, let us solve for the roots:

3x² + 15x + 5 = 0

Using the quadratic formula:

x = (-15 ± √(15² - 4*3*5))/(2*3)
x = (-15 ± √(225 - 60))/6
x = (-15 ± √165)/6

So we have 3x² + 15x + 5 = 3(x - (-15 + √165)/6)(x - (-15 - √165)/6)

Check:

3(x - (-15 + √165)/6)(x - (-15 - √165)/6)
3/36 (6x - (-15 + √165))(6x - (-15 - √165))
1/12 (6x - √165 + 15)(6x + √165 + 15)
1/12 (36x² + 6x√165 + 90x - 6x√165 - 165 - 15√165 + 90x + 15√165 + 225)
1/12 (36x² + 90x - 165 + 90x + 225)
1/12 (36x² + 180x + 60)
3x² + 15x + 5 ✓

Answer 2

The first step in this kind of problem is to factor the coefficient of the first term (the number in front of it) in as many ways as possible. In this case there's only one way: 3 = 3 x 1

So you can write (3x + ) (1x + )

leaving the second number in each term blank for the time being.

Next, factor the constant term, 5 in this case. Again, there's only one way to factor it: 5 = 5 x 1.

So there are two cases you can try:

Case 1: (3x + 5)(x + 1) and
Case 2: (3x + 1)(x + 5)

Let's try Case 1 using FOIL:

Multiply first terms: (3x)(x) = 3x^2
Outer terms: (3x)(1) = 3x
Inner terms: (5)(x) = 5x
Last terms: (1)(5) = 5

Adding them up, we get 3x^2 + 8x + 5, which isn't right.

Using the same technique for Case 2, we get

3x^2 + 16x + 5

which is also wrong. So this polynomial can't be factored.

Answer 3

actually, that one is completely factored...there's nothing to simplify any farther

You can't add 3x^2+15x because you can't add x^2+x. the base is the same but since you're ADDING, you can't do it.

i still have trouble with this....wait till u get to multiplying and dividing(First Outside Inside Last for a squared binomial)

And, yes, you did spell coefficient correctly

Answer 4

3x^2+15x+5
Discriminant = 15^2 -4(5)(3) =165
x1 = (-15 + sqrt(165) )/6
x2 = (-15 - sqrt(165) ) /6

3x^2+15x+5 =
3(x +(15 - sqrt(165))/6 )(x +(15 + sqrt(165))/6 )