Alright, let's go through a step-by-step factorization of the polynomial [tex]\( 5x^3 + 35x^2 + 6x + 42 \)[/tex].
The polynomial is given as:
[tex]\[ 5x^3 + 35x^2 + 6x + 42 \][/tex]
First, identify the task. We need to factor this polynomial completely and find the values of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] in the form [tex]\((5x^2 + A)(x + B)\)[/tex].
We can start by noting that [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants. Thus, if we multiply [tex]\((5x^2 + A)\)[/tex] by [tex]\((x + B)\)[/tex], it should yield the original polynomial. Let's distribute [tex]\((x + B)\)[/tex] through [tex]\((5x^2 + A)\)[/tex]:
[tex]\[
(5x^2 + A)(x + B) = 5x^2 \cdot x + 5x^2 \cdot B + A \cdot x + A \cdot B
\][/tex]
Expanding this, we get:
[tex]\[
= 5x^3 + 5Bx^2 + Ax + AB
\][/tex]
Now, let's align this with the original polynomial terms. This should equal:
[tex]\[
5x^3 + 35x^2 + 6x + 42
\][/tex]
By comparing the coefficients, we match:
1. The coefficient of [tex]\(x^3\)[/tex] is already 5, which matches with [tex]\(5\)[/tex].
2. The coefficient of [tex]\(x^2\)[/tex] is given by [tex]\(5B = 35\)[/tex]. Solving for [tex]\(B\)[/tex]:
[tex]\[
5B = 35 \implies B = 7
\][/tex]
3. The coefficient of [tex]\(x\)[/tex] is [tex]\(A\)[/tex], which is supposed to be [tex]\(6\)[/tex], so:
[tex]\[
A = 6
\][/tex]
4. Finally, [tex]\(AB\)[/tex] should match the constant term 42:
[tex]\[
(6)(7) = 42
\][/tex]
So, we have successfully matched all terms, and the values are consistent. Thus, the value of [tex]\(A\)[/tex] is:
[tex]\[ \boxed{6} \][/tex]
