To determine the largest possible value of the constant [tex]\( c \)[/tex] such that [tex]\( x + c \)[/tex] is a factor of the polynomial [tex]\( 3x^3 + 15x^2 - 378x \)[/tex], we need to follow a systematic approach.
1. Identify the polynomial:
[tex]\[
P(x) = 3x^3 + 15x^2 - 378x
\][/tex]
2. Factor out common terms:
This polynomial has a common factor of [tex]\( 3x \)[/tex] across all terms, so we can factor it out first:
[tex]\[
P(x) = 3x(x^2 + 5x - 126)
\][/tex]
3. Factor the quadratic expression:
Now, we need to factor the quadratic polynomial [tex]\( x^2 + 5x - 126 \)[/tex]. To do so, we look for two numbers that multiply to [tex]\(-126\)[/tex] and add up to [tex]\(5\)[/tex].
- The product is [tex]\( -126 \)[/tex]
- The sum is [tex]\( 5 \)[/tex]
These numbers are [tex]\( 14 \)[/tex] and [tex]\( -9 \)[/tex] because:
[tex]\[
14 \times (-9) = -126 \quad \text{and} \quad 14 + (-9) = 5
\][/tex]
4. Rewrite the quadratic polynomial:
[tex]\[
x^2 + 5x - 126 = (x + 14)(x - 9)
\][/tex]
5. Combine the factored form:
We substitute back into our original polynomial:
[tex]\[
P(x) = 3x(x + 14)(x - 9)
\][/tex]
6. Identify the factors:
The factors are [tex]\( 3x \)[/tex], [tex]\( (x + 14) \)[/tex], and [tex]\( (x - 9) \)[/tex]. We need [tex]\( x + c \)[/tex] to be one of the factors.
- For [tex]\( x + c \)[/tex] to be [tex]\( x + 14 \)[/tex], the value of [tex]\( c \)[/tex] is [tex]\( 14 \)[/tex].
- For [tex]\( x + c \)[/tex] to be [tex]\( x - 9 \)[/tex], the value of [tex]\( c \)[/tex] is [tex]\( -9 \)[/tex].
7. Find the largest possible value of [tex]\( c \)[/tex]:
The possible values for [tex]\( c \)[/tex] are [tex]\( 14 \)[/tex] and [tex]\( -9 \)[/tex]. The largest of these values is:
[tex]\[
c = 14
\][/tex]
Conclusion:
The largest possible value of the constant [tex]\( c \)[/tex] such that [tex]\( x + c \)[/tex] is a factor of [tex]\( 3x^3 + 15x^2 - 378x \)[/tex] is [tex]\( \boxed{14} \)[/tex].
