I have 2 questions about factoring polynomials?

Question

1st question:

How would you solve this?

4x^2 + 16x + 15

2nd question:

How would you solve this?

2x^3 + 4x^2 - 16x

Thanks in advance!!

Answer 1

the first one. it's always best to solve using the foil method when you have X^2.

with the first problem, 4 doesn't not divide easily into 15.

Since it's likely not to be solved with the foil method, just switch to the quadratic formula. It's really your last hope

the second problem.

you can easily pull out a 2x. so the problem becomes 2x (x^2+2x-8)

you can use the foil method easily on this problem roots of 2 and 4.

Answer 2

In the first I would use the quadratic formula. Sometimes the answer just pops out at me, but in this one, since there's a 4x^2 rather than an x^2 it's harder to directly work out the answer, so the quadratic formula is easier.

I like to pull out the squared factor also, since it simplifies the result, so a=1

4x^2 +16x +15

= 4(x^2 + 4x - 15/4)

factorize x^2 + 4x - 15/4 with the quadratic formula

x = (-4 +- sqrt(16 - 4*-15/4) )/2

= (-4 +- sqrt(16 + 15))/2

= -2 + sqrt(31)/2 and -2 - sqrt(31)/2

[here I use the calculator, since sqrt(31) is not a whole number]

so the expression is zero when x = 0.784 or -4.784 (approx)

So (x - 0.784)(x + 4.784)

So adding back the 4 we took out first the factorization is

4(x-0.784)(x+0.784)

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In the second, first take out the 2x to yield (2x)(some quadratic). Then use the quadratic formula if the answer isn't obvious.

2x^3+ 4x^2 -16x = (2x)( x^2 + 2x - 8)

Using the quadratic formula on the quadratic bit:

x= (-2 +- sqrt(4 - 4*-8)) / 2

= (-2 +- sqrt(36))/2

= (-2 + 6)/2 and (-2 - 6)/2

= 2 and -4

So the factorization is (2x)(x-2)(x+4)

With a little practice and/or thought, you won't need the quadratic formula to factorize (x^2 + 2x - 8)

You know it's going to be (x + a)(x + b)

For a and b, think of values for them, where a+b = 2 and a*b = -8.

With a little practice, it is easy to see those numbers are +4 and -2. Hence (x -2)(x+4).