explain in your own words how to factor a polynomial of the the form ax^2 + bx+c when A is not equal to 1. Describe both the grouping approch as well as reversing FOIL. Contrast the two methods by means of an example. Which is the best approach and why>
"explain in your own words "
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This is the question I am trying to answer
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ax^2+bx+c= ax^2(bx+c)
ax^2+bx+c= a(x^2+bx+c)
I don't understand it, I don't even know if I am on the correct track
Factoring a polynomial of the form ax^2 + bx + c, where a is not equal to 1, can be done using two different approaches: the grouping approach and reversing FOIL. Let's go through each method and then compare them with an example.
1. Grouping Approach:
a. Write down the polynomial in the form of two pairs:
ax^2 + bx + c = (ax^2 + mx) + (nx + c)
b. Factor out the greatest common factor (GCF) from each pair:
ax^2 + mx becomes x(a + m)
nx + c becomes c + n
c. Combine the factored pairs:
(ax^2 + mx) + (nx + c) = x(a + m) + (c + n)
d. Factor out the GCF from the resulting expression:
x(a + m) + (c + n) = x(a + m) + 1(c + n)
e. Factor out the common binomial factor:
x(a + m) + 1(c + n) = (a + m)x + (c + n)
2. Reversing FOIL:
a. Write down the polynomial in the form of two binomials:
ax^2 + bx + c = (px + q)(rx + s)
b. Expand the binomial multiplication using the FOIL method:
(px + q)(rx + s) = (p)(rx) + (p)(s) + (q)(rx) + (q)(s)
c. Simplify and combine like terms:
pxr + ps + qrx + qs
d. Match the expanded expression with the original polynomial:
pxr + ps + qrx + qs = ax^2 + bx + c
e. Equate the corresponding coefficients:
pxr = a (1)
ps + qrx = b (2)
qs = c (3)
f. Solve equations (1), (2), and (3) simultaneously to find the values of p, q, r, and s.
Now let's compare these two approaches using an example:
Example: Factor the polynomial 2x^2 + 7x + 3.
Grouping Approach:
(2x^2 + 3x) + (4x + 3)
x(2x + 3) + 1(4x + 3)
(2x + 3)x + (4x + 3)
Reversing FOIL:
(2x + 1)(x + 3)
2x^2 + 6x + x + 3
2x^2 + 7x + 3
The best approach depends on the specific polynomial being factored and personal preference. In general, the grouping approach is simpler and often more straightforward. It is particularly useful when the polynomial has common factors among its terms. However, the reversing FOIL approach can be beneficial when the polynomial is not easily factored using common factors and requires a more algebraic approach. Ultimately, it is good to be familiar with both methods and choose the one that suits the given polynomial best.