Answer :
To factor the polynomial [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] completely, we can follow these steps:
1. Look for Common Factors: At first glance, the terms [tex]\( x^3 \)[/tex], [tex]\( 3x^2 \)[/tex], [tex]\( 16x \)[/tex], and [tex]\( 48 \)[/tex] don't have any obvious common factors among all terms, other than 1. Thus, the polynomial does not have a common factor we can factor out initially.
2. Apply the Factor Theorem: To determine possible factors of the polynomial, test small integer values such as [tex]\( \pm 1, \pm 2, \pm 3, \)[/tex] etc., as potential roots.
3. Identify Potential Roots:
- Let's test [tex]\( x = -3 \)[/tex] to see if it is a root.
- Substitute [tex]\( x = -3 \)[/tex] into the polynomial:
[tex]\[
(-3)^3 + 3(-3)^2 + 16(-3) + 48 = -27 + 27 - 48 + 48 = 0
\][/tex]
Since the polynomial equals zero, [tex]\( x = -3 \)[/tex] is a root. Therefore, [tex]\( x + 3 \)[/tex] is a factor.
4. Polynomial Division: Now that [tex]\( x + 3 \)[/tex] is a factor, perform polynomial division of [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] by [tex]\( x + 3 \)[/tex].
- Divide [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] by [tex]\( x + 3 \)[/tex] using synthetic or long division. After the division, you'll obtain:
[tex]\[
x^3 + 3x^2 + 16x + 48 = (x + 3)(x^2 + 16)
\][/tex]
5. Factor Completely: Now, explore the quadratic factor:
- The quadratic [tex]\( x^2 + 16 \)[/tex] does not factor further using real numbers, because it has no real roots (the discriminant [tex]\( b^2 - 4ac = 0 - 4(1)(16) = -64 \)[/tex] is negative).
6. Final Factored Form: Thus, the polynomial is completely factored as:
[tex]\[
(x + 3)(x^2 + 16)
\][/tex]
This is the complete factorization of the polynomial [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex].
1. Look for Common Factors: At first glance, the terms [tex]\( x^3 \)[/tex], [tex]\( 3x^2 \)[/tex], [tex]\( 16x \)[/tex], and [tex]\( 48 \)[/tex] don't have any obvious common factors among all terms, other than 1. Thus, the polynomial does not have a common factor we can factor out initially.
2. Apply the Factor Theorem: To determine possible factors of the polynomial, test small integer values such as [tex]\( \pm 1, \pm 2, \pm 3, \)[/tex] etc., as potential roots.
3. Identify Potential Roots:
- Let's test [tex]\( x = -3 \)[/tex] to see if it is a root.
- Substitute [tex]\( x = -3 \)[/tex] into the polynomial:
[tex]\[
(-3)^3 + 3(-3)^2 + 16(-3) + 48 = -27 + 27 - 48 + 48 = 0
\][/tex]
Since the polynomial equals zero, [tex]\( x = -3 \)[/tex] is a root. Therefore, [tex]\( x + 3 \)[/tex] is a factor.
4. Polynomial Division: Now that [tex]\( x + 3 \)[/tex] is a factor, perform polynomial division of [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] by [tex]\( x + 3 \)[/tex].
- Divide [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex] by [tex]\( x + 3 \)[/tex] using synthetic or long division. After the division, you'll obtain:
[tex]\[
x^3 + 3x^2 + 16x + 48 = (x + 3)(x^2 + 16)
\][/tex]
5. Factor Completely: Now, explore the quadratic factor:
- The quadratic [tex]\( x^2 + 16 \)[/tex] does not factor further using real numbers, because it has no real roots (the discriminant [tex]\( b^2 - 4ac = 0 - 4(1)(16) = -64 \)[/tex] is negative).
6. Final Factored Form: Thus, the polynomial is completely factored as:
[tex]\[
(x + 3)(x^2 + 16)
\][/tex]
This is the complete factorization of the polynomial [tex]\( x^3 + 3x^2 + 16x + 48 \)[/tex].
