Answer :
Sure! Let's factorize the expression [tex]\( x^4 - 4x^3 - 21x^2 \)[/tex] step-by-step.
### Step 1: Identify the common factor
First, identify the common factor in the expression. Notice that each term has at least [tex]\( x^2 \)[/tex] as a factor.
[tex]\[ x^4 - 4x^3 - 21x^2 = x^2 (x^2 - 4x - 21) \][/tex]
### Step 2: Factor the quadratic
Now, we need to factorize the quadratic expression [tex]\( x^2 - 4x - 21 \)[/tex].
To factorize this quadratic expression, we need to find two numbers that multiply to the constant term ([tex]\(-21\)[/tex]) and add up to the coefficient of the linear term ([tex]\(-4\)[/tex]).
The two numbers that satisfy these requirements are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex], because:
[tex]\[ -7 + 3 = -4 \][/tex]
[tex]\[ -7 \cdot 3 = -21 \][/tex]
So, we can write:
[tex]\[ x^2 - 4x - 21 = (x - 7)(x + 3) \][/tex]
### Step 3: Combine the results
Now we combine the common factor with the factored quadratic expression:
[tex]\[ x^4 - 4x^3 - 21x^2 = x^2 (x - 7)(x + 3) \][/tex]
### Conclusion
Thus, the factorized form of the expression [tex]\( x^4 - 4x^3 - 21x^2 \)[/tex] is:
[tex]\[ x^2 (x - 7)(x + 3) \][/tex]
This is the final factored form of the given expression.
### Step 1: Identify the common factor
First, identify the common factor in the expression. Notice that each term has at least [tex]\( x^2 \)[/tex] as a factor.
[tex]\[ x^4 - 4x^3 - 21x^2 = x^2 (x^2 - 4x - 21) \][/tex]
### Step 2: Factor the quadratic
Now, we need to factorize the quadratic expression [tex]\( x^2 - 4x - 21 \)[/tex].
To factorize this quadratic expression, we need to find two numbers that multiply to the constant term ([tex]\(-21\)[/tex]) and add up to the coefficient of the linear term ([tex]\(-4\)[/tex]).
The two numbers that satisfy these requirements are [tex]\(-7\)[/tex] and [tex]\(3\)[/tex], because:
[tex]\[ -7 + 3 = -4 \][/tex]
[tex]\[ -7 \cdot 3 = -21 \][/tex]
So, we can write:
[tex]\[ x^2 - 4x - 21 = (x - 7)(x + 3) \][/tex]
### Step 3: Combine the results
Now we combine the common factor with the factored quadratic expression:
[tex]\[ x^4 - 4x^3 - 21x^2 = x^2 (x - 7)(x + 3) \][/tex]
### Conclusion
Thus, the factorized form of the expression [tex]\( x^4 - 4x^3 - 21x^2 \)[/tex] is:
[tex]\[ x^2 (x - 7)(x + 3) \][/tex]
This is the final factored form of the given expression.
