Answer :
To factor the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for any common factors in all the terms of the polynomial.
In [tex]\( x^3 - 7x^2 + 10x \)[/tex], each term has at least one [tex]\( x \)[/tex].
So, the GCF is [tex]\( x \)[/tex].
Factor out the GCF:
[tex]\[
x(x^2 - 7x + 10)
\][/tex]
2. Factor the Quadratic Expression:
Now, focus on the quadratic [tex]\( x^2 - 7x + 10 \)[/tex].
Look for two numbers that multiply to [tex]\( 10 \)[/tex] (the constant term) and add up to [tex]\(-7\)[/tex] (the coefficient of the linear term):
The numbers [tex]\(-5\)[/tex] and [tex]\(-2\)[/tex] fit this requirement because:
[tex]\[
(-5) \times (-2) = 10 \quad \text{and} \quad (-5) + (-2) = -7
\][/tex]
Therefore, factor the quadratic as:
[tex]\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\][/tex]
3. Combine the Factors:
Now, substitute back the factors of the quadratic into the expression we factored out earlier:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] is:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
This shows the polynomial broken down into linear factors, revealing its roots and simplifying the expression.
1. Identify the Greatest Common Factor (GCF):
First, look for any common factors in all the terms of the polynomial.
In [tex]\( x^3 - 7x^2 + 10x \)[/tex], each term has at least one [tex]\( x \)[/tex].
So, the GCF is [tex]\( x \)[/tex].
Factor out the GCF:
[tex]\[
x(x^2 - 7x + 10)
\][/tex]
2. Factor the Quadratic Expression:
Now, focus on the quadratic [tex]\( x^2 - 7x + 10 \)[/tex].
Look for two numbers that multiply to [tex]\( 10 \)[/tex] (the constant term) and add up to [tex]\(-7\)[/tex] (the coefficient of the linear term):
The numbers [tex]\(-5\)[/tex] and [tex]\(-2\)[/tex] fit this requirement because:
[tex]\[
(-5) \times (-2) = 10 \quad \text{and} \quad (-5) + (-2) = -7
\][/tex]
Therefore, factor the quadratic as:
[tex]\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\][/tex]
3. Combine the Factors:
Now, substitute back the factors of the quadratic into the expression we factored out earlier:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] is:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
This shows the polynomial broken down into linear factors, revealing its roots and simplifying the expression.
