Sure, let's factor the quadratic polynomial [tex]\( d^2 + 49d - 50 \)[/tex].
To factor the polynomial [tex]\( d^2 + 49d - 50 \)[/tex], we need to find two binomials [tex]\((d + a)(d + b)\)[/tex] such that when expanded, they give the original polynomial. This means:
[tex]\[
d^2 + 49d - 50 = (d + a)(d + b)
\][/tex]
When we expand the right-hand side, we get:
[tex]\[
(d + a)(d + b) = d^2 + (a + b)d + ab
\][/tex]
By comparing coefficients, we need:
1. [tex]\(a + b = 49\)[/tex] (the coefficient of [tex]\(d\)[/tex])
2. [tex]\(ab = -50\)[/tex] (the constant term)
Now, let's find two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy these requirements.
The numbers 50 and -1 satisfy both conditions:
1. [tex]\(50 + (-1) = 49\)[/tex]
2. [tex]\(50 \cdot (-1) = -50\)[/tex]
So, we can write the quadratic polynomial as:
[tex]\[
d^2 + 49d - 50 = (d - 1)(d + 50)
\][/tex]
Thus, the factorization of [tex]\(d^2 + 49d - 50\)[/tex] is:
[tex]\[
(d - 1)(d + 50)
\][/tex]
