Certainly! Let's factor the quadratic polynomial [tex]\( u^2 + 49u + 94 \)[/tex] step by step.
1. Identify the quadratic polynomial:
The polynomial we are working with is [tex]\( u^2 + 49u + 94 \)[/tex].
2. Look for a form:
We want to factor this into the form [tex]\( (u + a)(u + b) \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are numbers we need to find.
3. Apply factoring:
We need to find two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[
a + b = 49
\][/tex]
and
[tex]\[
ab = 94
\][/tex]
4. Determine the factors:
Solving the above equations for the numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we get [tex]\( a = 2 \)[/tex] and [tex]\( b = 47 \)[/tex].
5. Verification:
To verify, we check if [tex]\( (u + 2)(u + 47) \)[/tex] expands back to [tex]\( u^2 + 49u + 94 \)[/tex]:
[tex]\[
(u + 2)(u + 47) = u^2 + 47u + 2u + 94 = u^2 + 49u + 94
\][/tex]
This confirms that the factors are correct.
Therefore, the factored form of the polynomial [tex]\( u^2 + 49u + 94 \)[/tex] is
[tex]\[
(u + 2)(u + 47).
\][/tex]
