Factor the given polynomial completely:
1. 2px^2 + 7px +3p
2. 24a^2 - 150b^2
1. 2px^2 + 7px +3p
2. 24a^2 - 150b^2
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1. 2px^2 + 7px +3p
Factor out the common p term first:
p(2x^2 + 7x + 3)
The 2x^2 term requires that 2x be in one factor and x in the other:
(2x +- ?)(x +- ?)
Since the 3 is positive, the signs in the factors are the same. Since the 7x is positive, they're both plus signs:
(2x + ?)(x + ?)
Now you need two factors of 3 that add to 7 when one of them is multiplied by 2. The only integer factors of 3 are 1 and 3. Arrange them so that the 3 gets multiplied by the 2x:
(2x + 1)(x + 3)
Don't forget the p term:
p(2x + 1)(x + 3)
2. 24a^2 - 150b^2
Start by factoring out 2:
2(12a^2 - 75b^2)
The 12 and 75 have a common factor of 3. Factor that out:
6(4a^2 - 25b^2)
The 4a^2 - 25b^2 is a "difference of two squares" factoring.
The formula for that is a^2 - b^2 = (a + b)(a - b)
Substitute 2a for a and 5b for b:
6(2a + 5b)(2a - 5b)
Factor out the common p term first:
p(2x^2 + 7x + 3)
The 2x^2 term requires that 2x be in one factor and x in the other:
(2x +- ?)(x +- ?)
Since the 3 is positive, the signs in the factors are the same. Since the 7x is positive, they're both plus signs:
(2x + ?)(x + ?)
Now you need two factors of 3 that add to 7 when one of them is multiplied by 2. The only integer factors of 3 are 1 and 3. Arrange them so that the 3 gets multiplied by the 2x:
(2x + 1)(x + 3)
Don't forget the p term:
p(2x + 1)(x + 3)
2. 24a^2 - 150b^2
Start by factoring out 2:
2(12a^2 - 75b^2)
The 12 and 75 have a common factor of 3. Factor that out:
6(4a^2 - 25b^2)
The 4a^2 - 25b^2 is a "difference of two squares" factoring.
The formula for that is a^2 - b^2 = (a + b)(a - b)
Substitute 2a for a and 5b for b:
6(2a + 5b)(2a - 5b)