Answer :
- Factor out the common factor $x^2$: $x^4 - 23x^3 + 126x^2 = x^2(x^2 - 23x + 126)$.
- Find two numbers that multiply to 126 and add up to -23: -9 and -14.
- Rewrite the quadratic expression as $(x - 9)(x - 14)$.
- The completely factored expression is $\boxed{x^2(x - 9)(x - 14)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the expression $x^4 - 23x^3 + 126x^2$ completely. This means we want to write it as a product of simpler expressions.
2. Factoring out the Common Factor
First, notice that each term in the expression has a factor of $x^2$. We can factor out $x^2$ from the entire expression: $$x^4 - 23x^3 + 126x^2 = x^2(x^2 - 23x + 126)$$
3. Factoring the Quadratic Expression
Now we need to factor the quadratic expression $x^2 - 23x + 126$. We are looking for two numbers that multiply to 126 and add up to -23. Let's list the factor pairs of 126:
1 and 126
2 and 63
3 and 42
6 and 21
7 and 18
9 and 14
Since we need the two numbers to add up to -23, we consider the negative pairs. The pair -9 and -14 satisfies the condition since $(-9) \times (-14) = 126$ and $(-9) + (-14) = -23$.
4. Writing the Factored Expression
So we can rewrite the quadratic expression as $(x - 9)(x - 14)$. Therefore, the completely factored expression is: $$x^2(x - 9)(x - 14)$$
5. Final Answer
The completely factored expression is $x^2(x - 9)(x - 14)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and science. For example, in physics, you might need to factor a polynomial to find the roots of an equation that describes the motion of an object. In engineering, factoring can help simplify complex expressions when designing structures or circuits. In computer science, factoring is used in cryptography and data compression algorithms.
- Find two numbers that multiply to 126 and add up to -23: -9 and -14.
- Rewrite the quadratic expression as $(x - 9)(x - 14)$.
- The completely factored expression is $\boxed{x^2(x - 9)(x - 14)}$.
### Explanation
1. Understanding the Problem
We are asked to factor the expression $x^4 - 23x^3 + 126x^2$ completely. This means we want to write it as a product of simpler expressions.
2. Factoring out the Common Factor
First, notice that each term in the expression has a factor of $x^2$. We can factor out $x^2$ from the entire expression: $$x^4 - 23x^3 + 126x^2 = x^2(x^2 - 23x + 126)$$
3. Factoring the Quadratic Expression
Now we need to factor the quadratic expression $x^2 - 23x + 126$. We are looking for two numbers that multiply to 126 and add up to -23. Let's list the factor pairs of 126:
1 and 126
2 and 63
3 and 42
6 and 21
7 and 18
9 and 14
Since we need the two numbers to add up to -23, we consider the negative pairs. The pair -9 and -14 satisfies the condition since $(-9) \times (-14) = 126$ and $(-9) + (-14) = -23$.
4. Writing the Factored Expression
So we can rewrite the quadratic expression as $(x - 9)(x - 14)$. Therefore, the completely factored expression is: $$x^2(x - 9)(x - 14)$$
5. Final Answer
The completely factored expression is $x^2(x - 9)(x - 14)$.
### Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and science. For example, in physics, you might need to factor a polynomial to find the roots of an equation that describes the motion of an object. In engineering, factoring can help simplify complex expressions when designing structures or circuits. In computer science, factoring is used in cryptography and data compression algorithms.
