Answer :
To express the function [tex]\( y = 2x^3 - 10x^2 - 4x + 48 \)[/tex] in factored form, follow these steps:
1. Understand the Problem: We need to find the factored form of the cubic polynomial. To do this, we need to identify its roots, which are the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex].
2. Find the Roots of the Polynomial:
- The polynomial can be factored by finding the values of [tex]\( x \)[/tex] where the polynomial equals zero.
- For a cubic polynomial, there can be a combination of real and/or complex roots, but given that a graph helps us, let's assume we have real roots to simplify the problem.
3. Assessing the Equation:
- The graph indicates the locations where the polynomial crosses or touches the x-axis. These x-values are the roots.
4. Factoring the Polynomial:
- Assuming the graph reveals roots at [tex]\( x = 4 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -2 \)[/tex], we can express the polynomial as a product of its factors based on these roots:
[tex]\[
y = 2(x - 4)(x - 3)(x + 2)
\][/tex]
5. Double-Check:
- Make sure the factorized form expands back to the original polynomial, confirming if possible. Here, it should check out correctly, matching the original coefficients upon expansion.
Thus, the factored form of the function is [tex]\( y = 2(x - 4)(x - 3)(x + 2) \)[/tex].
1. Understand the Problem: We need to find the factored form of the cubic polynomial. To do this, we need to identify its roots, which are the values of [tex]\( x \)[/tex] for which [tex]\( y = 0 \)[/tex].
2. Find the Roots of the Polynomial:
- The polynomial can be factored by finding the values of [tex]\( x \)[/tex] where the polynomial equals zero.
- For a cubic polynomial, there can be a combination of real and/or complex roots, but given that a graph helps us, let's assume we have real roots to simplify the problem.
3. Assessing the Equation:
- The graph indicates the locations where the polynomial crosses or touches the x-axis. These x-values are the roots.
4. Factoring the Polynomial:
- Assuming the graph reveals roots at [tex]\( x = 4 \)[/tex], [tex]\( x = 3 \)[/tex], and [tex]\( x = -2 \)[/tex], we can express the polynomial as a product of its factors based on these roots:
[tex]\[
y = 2(x - 4)(x - 3)(x + 2)
\][/tex]
5. Double-Check:
- Make sure the factorized form expands back to the original polynomial, confirming if possible. Here, it should check out correctly, matching the original coefficients upon expansion.
Thus, the factored form of the function is [tex]\( y = 2(x - 4)(x - 3)(x + 2) \)[/tex].
