Worksheet For Quadratics With A Coefficient Greater Than 1 printable pdf download

Worksheet For Quadratics With A Coefficient Greater Than 1

Homework 3

Amneris suggested the following factoring algorithm for quadratics with

a coeﬃcient greater than 1 on the x

term as in the following example.

Suppose we want to factor (2x

2). Multiply the coeﬃcient of x

and

the constant term to get

4. We then look for factors of

4 that add to

In this case, the factors are

4 and 1. We now write

4)(x + 1).

Since a was 2, we check to see which constant term 2 factors. In this case, 2

is a factor of

4. We then divide

4 by 2 and put the 2 in front of the x in

(x + 1) to get

2 = (x

2)(2x + 1).

Often the process of ﬁnding the factors is done with the following graphic:

1. Explain why this method works. (Your explanation should include

a proof that the method works, not simply be a “heuristic” reason

why you might do this. However, it should also include a heuristic

explanation.)

There are two things to check. First, whether if the method

is successful that what arises is a solution, and second that if the

method is not successful then there is no factorization. We begin with

the ﬁrst.

Suppose we are given the function ax

+ bx + c = 0 and s

and s

are integers such that s

= ac and s

+ s

= b. Moreover, assume

a s

. Then s

= a r

for some integer r

. Now the method gives the

factorization (x + r

)(ax + s

) for the polynomial. Multiplying this out

Worksheet For Quadratics With A Coefficient Greater Than 1

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