In this lesson, you will learn to factor trinomials in which the coefficient of x2 is a number other than 1.
The FOIL method will help you factor trinomials without
models.
Factor each trinomial.
Alternative Solutions:
2x2 is the product
of the First terms, and 3 is the product of the Last terms.
The last term, 3, is positive. The
sum of the inside and outside terms, –7, is negative. So, both factors
of 3 must be negative. Try factor pairs of 3 until the sum of the products of
the Outer and Inner terms is –7x.
Therefore, 2x2 – 7x
+ 3 = (2x – 1)(x – 3).
2.
3y2 + 2y – 5
Alternative Solutions:
3y2
is the product of the First terms, and –5 is
the product of the Last terms.
Find
integers whose product is –5. Try factor
pairs of –5 until the sum of the products of the Outer and Inner terms
is 2y.
Therefore,
3y2 + 2y – 5 = (3y
+ 5)(y – 1).
Sometimes the coefficient of x2 can be
factored into more than one pair of integers.
Example
3.
Factor 4x2 + 12x + 5.
Alternative Solutions:
Therefore, 4x2
+ 12x + 5 = (2x + 5)(2x + 1).
Recall that the first step in factoring any polynomial is to
factor out any GCF other than 1.
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4. The volume of a rectangular shipping crate is 6x3
– 15x2 – 36x. Find possible dimensions for the
crate.
Alternative Solutions:
The
formula for the volume of a rectangular prism is V = lwh. Find three
factors of 6x3 – 15x2
– 36x. First, look for a GCF.
3x
is one factor of 6x3 – 15x2
– 36x. Factor 2x2 – 5x – 12 to find the other
two factors.
The
factors of –12 are –3 and 4, 3 and –4, –2 and 6, 2 and –6, –1 and 12, and 1 and
–12. Check several combinations; the correct factors are 3 and –4.
2x2
– 5x – 12 = (2x + 3)(x – 4)
So, 6x3
– 15x2 – 36x = 3x(2x + 3)(x – 4).
Therefore, the dimensions can be 3x, 2x + 3, and x – 4.
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