Answer :
- Identify the greatest common factor (GCF) of the terms in the expression.
- Factor out the GCF from each term.
- Write the expression in the form GCF * (remaining expression).
- The factored expression is $\boxed{x^3}$
### Explanation
1. Understanding the Problem
We are asked to factor out the greatest common factor (GCF) from the expression $x^9 + 4x^7 - 11x^3$. The GCF is the largest expression that divides each term of the polynomial.
2. Finding the Greatest Common Factor
Let's identify the GCF of the terms $x^9$, $4x^7$, and $-11x^3$. The GCF will be a power of $x$. We look for the smallest exponent of $x$ that appears in all terms. The exponents are 9, 7, and 3. The smallest of these is 3. Therefore, the GCF is $x^3$.
3. Factoring out the GCF
Now, we factor out $x^3$ from each term in the expression:
$x^9 = x^3 \cdot x^6$
$4x^7 = x^3 \cdot 4x^4$
$-11x^3 = x^3 \cdot (-11)$
4. Writing the Factored Expression
So, we can rewrite the expression as:
$x^9 + 4x^7 - 11x^3 = x^3(x^6) + x^3(4x^4) + x^3(-11) = x^3(x^6 + 4x^4 - 11)$.
5. Final Answer
Thus, the factored expression is $x^3(x^6 + 4x^4 - 11)$.
### Examples
Factoring out the greatest common factor is a fundamental skill in algebra. It's like simplifying a recipe: if all ingredients have a common unit (like 'cups'), you can factor that out to make the recipe easier to read and understand. In real life, this can be used to simplify complex formulas in physics or engineering, making calculations easier and more efficient. For example, if you're calculating the total cost of items with a common discount factor, factoring out the discount simplifies the calculation.
- Factor out the GCF from each term.
- Write the expression in the form GCF * (remaining expression).
- The factored expression is $\boxed{x^3}$
### Explanation
1. Understanding the Problem
We are asked to factor out the greatest common factor (GCF) from the expression $x^9 + 4x^7 - 11x^3$. The GCF is the largest expression that divides each term of the polynomial.
2. Finding the Greatest Common Factor
Let's identify the GCF of the terms $x^9$, $4x^7$, and $-11x^3$. The GCF will be a power of $x$. We look for the smallest exponent of $x$ that appears in all terms. The exponents are 9, 7, and 3. The smallest of these is 3. Therefore, the GCF is $x^3$.
3. Factoring out the GCF
Now, we factor out $x^3$ from each term in the expression:
$x^9 = x^3 \cdot x^6$
$4x^7 = x^3 \cdot 4x^4$
$-11x^3 = x^3 \cdot (-11)$
4. Writing the Factored Expression
So, we can rewrite the expression as:
$x^9 + 4x^7 - 11x^3 = x^3(x^6) + x^3(4x^4) + x^3(-11) = x^3(x^6 + 4x^4 - 11)$.
5. Final Answer
Thus, the factored expression is $x^3(x^6 + 4x^4 - 11)$.
### Examples
Factoring out the greatest common factor is a fundamental skill in algebra. It's like simplifying a recipe: if all ingredients have a common unit (like 'cups'), you can factor that out to make the recipe easier to read and understand. In real life, this can be used to simplify complex formulas in physics or engineering, making calculations easier and more efficient. For example, if you're calculating the total cost of items with a common discount factor, factoring out the discount simplifies the calculation.
