How to factor by grouping?

Question

How would you go about factoring a polynomial such as x³+5x+15+3x²?

Answer 1

First group the terms so that the difference in the exponents in the first two terms is the same as the difference in the exponents in the second two terms:

x³+5x+15+3x² → x³+3x²+5x+15 (the difference between x³ & x² is is a power of one and the difference between 5x & 15 is also a power of one)

Then factor out the greatest common divisor from the first two terms:

x³+3x²+5x+15 → (x²)(x+3)+5x+15 (the largest common divisor of x³ & 3x² is x²)

Repeat with the second two terms:

(x²)(x+3)+5x+15 → (x²)(x+3)+(5)(x+3) (the largest common divisor of 5x & 15 is 5)

This should leave the remaining binomials the same (x+3) & (x+3) they are the same but with opposite signs (x-3) & (-x+3) simply factor out a negative from one and put it in front of the largest common divisor from the previous step

Then combine the multiplier of the first binomial with the multiplier of the second binomial (this is equivalent to factoring out the binomial):

(x²)(x+3)+(5)(x+3)→ (x²+5)(x+3) (if the 5 was negative then it would be (x²-5)

(x²+5)(x+3) is the final answer!

Try and follow these steps with 35x³+ 20x² + 21x + 12 (the correct answer is below)

Correct Answer:

(5x²+3)(7x+4)

Answer 2

First, put it into standard form:

x³ + 3x² + 5x + 15

Parentheses around the first two terms and the last two terms:

(x³ + 3x²)+ (5x + 15)

Factor the greatest common factor out of each:

x²(x+3) + 5(x+3)

Factor out the x+3:

(x+3)(x²+5)

If you have an x² factor, check to see if it can be further factored (as a quadratic). In this case, the x² + 5 cannot be further factored, so you're done!