If
$c$ is positive, then the factors of
$c$ must be either both positive or both negative. The factors are both negative if
$b$ is negative, and are both positive if
$b$ is positive. If
$c$ is negative, it means only one of the factors of
$c$ is negative, the other one being positive.
Once you get an answer, multiply out your brackets again just to make sure it really works.
Find the factors for the following trinomial expressions:
$2{x}^{2}+11x+5$
$3{x}^{2}+19x+6$
$6{x}^{2}+7x+2$
$12{x}^{2}+8x+1$
$8{x}^{2}+6x+1$
Find the factors for the following trinomials:
$3{x}^{2}+17x-6$
$7{x}^{2}-6x-1$
$8{x}^{2}-6x+1$
$2{x}^{2}-5x-3$
Factorisation by grouping
One other method of factorisation involves the use of common factors. We know that the factors of
$3x+3$ are 3 and
$(x+1)$ . Similarly, the factors of
$2{x}^{2}+2x$ are
$2x$ and
$(x+1)$ . Therefore, if we have an expression:
$$2{x}^{2}+2x+3x+3$$
then we can factorise as:
$$2x(x+1)+3(x+1).$$
You can see that there is another common factor:
$x+1$ . Therefore, we can now write:
$$(x+1)(2x+3).$$
We get this by taking out the
$x+1$ and seeing what is left over. We have a
$+2x$ from the first term and a
$+3$ from the second term. This is called
factorisation by grouping .
Find the factors of
$7x+14y+bx+2by$ by grouping
There are no factors that are common to all terms.
7 is a common factor of the first two terms and
$b$ is a common factor of the second two terms.
$$7x+14y+bx+2by=7(x+2y)+b(x+2y)$$
$x+2y$ is a common factor.
$$7(x+2y)+b(x+2y)=(x+2y)(7+b)$$
The factors of
$7x+14y+bx+2by$ are
$(7+b)$ and
$(x+2y)$ .
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