Answer :
To factor the polynomial
[tex]$$
x^4 + 11x^3 + 28x^2,
$$[/tex]
we begin by noticing a common factor in all terms.
Step 1. Factor Out the Greatest Common Factor
Every term in the polynomial contains at least [tex]$x^2$[/tex], so we factor [tex]$x^2$[/tex] out:
[tex]$$
x^4 + 11x^3 + 28x^2 = x^2 \left(x^2 + 11x + 28\right).
$$[/tex]
Step 2. Factor the Quadratic
Next, we factor the quadratic
[tex]$$
x^2 + 11x + 28.
$$[/tex]
We look for two numbers that multiply to [tex]$28$[/tex] (the constant term) and add to [tex]$11$[/tex] (the coefficient of [tex]$x$[/tex]). The numbers [tex]$4$[/tex] and [tex]$7$[/tex] satisfy these conditions because:
[tex]$$
4 \times 7 = 28 \quad \text{and} \quad 4 + 7 = 11.
$$[/tex]
Thus, the quadratic factors as:
[tex]$$
x^2 + 11x + 28 = (x + 4)(x + 7).
$$[/tex]
Step 3. Write the Complete Factorization
Substituting the factorization back, we get:
[tex]$$
x^4 + 11x^3 + 28x^2 = x^2 (x + 4)(x + 7).
$$[/tex]
Verification Using a Graphing Calculator
To verify, when you graph the function
[tex]$$
y = x^4 + 11x^3 + 28x^2,
$$[/tex]
you will observe zeros at:
- [tex]$x = 0$[/tex] (with multiplicity 2),
- [tex]$x = -4$[/tex], and
- [tex]$x = -7$[/tex].
These zeros confirm that the factorization is correct.
Thus, the final answer is:
[tex]$$
\boxed{x^4 + 11x^3 + 28x^2 = x^2 (x + 4)(x + 7)}.
$$[/tex]
This corresponds to Choice A: Factor completely.
[tex]$$
x^4 + 11x^3 + 28x^2,
$$[/tex]
we begin by noticing a common factor in all terms.
Step 1. Factor Out the Greatest Common Factor
Every term in the polynomial contains at least [tex]$x^2$[/tex], so we factor [tex]$x^2$[/tex] out:
[tex]$$
x^4 + 11x^3 + 28x^2 = x^2 \left(x^2 + 11x + 28\right).
$$[/tex]
Step 2. Factor the Quadratic
Next, we factor the quadratic
[tex]$$
x^2 + 11x + 28.
$$[/tex]
We look for two numbers that multiply to [tex]$28$[/tex] (the constant term) and add to [tex]$11$[/tex] (the coefficient of [tex]$x$[/tex]). The numbers [tex]$4$[/tex] and [tex]$7$[/tex] satisfy these conditions because:
[tex]$$
4 \times 7 = 28 \quad \text{and} \quad 4 + 7 = 11.
$$[/tex]
Thus, the quadratic factors as:
[tex]$$
x^2 + 11x + 28 = (x + 4)(x + 7).
$$[/tex]
Step 3. Write the Complete Factorization
Substituting the factorization back, we get:
[tex]$$
x^4 + 11x^3 + 28x^2 = x^2 (x + 4)(x + 7).
$$[/tex]
Verification Using a Graphing Calculator
To verify, when you graph the function
[tex]$$
y = x^4 + 11x^3 + 28x^2,
$$[/tex]
you will observe zeros at:
- [tex]$x = 0$[/tex] (with multiplicity 2),
- [tex]$x = -4$[/tex], and
- [tex]$x = -7$[/tex].
These zeros confirm that the factorization is correct.
Thus, the final answer is:
[tex]$$
\boxed{x^4 + 11x^3 + 28x^2 = x^2 (x + 4)(x + 7)}.
$$[/tex]
This corresponds to Choice A: Factor completely.
