Answer :
To factor the expression [tex]\( x^3 - 7x^2 + 10x \)[/tex] completely, follow these steps:
1. Look for common factors:
Start by identifying any common factors in the terms of the expression. Here, each term contains a factor of [tex]\( x \)[/tex]. So, factor [tex]\( x \)[/tex] out of the expression:
[tex]\[
x(x^2 - 7x + 10)
\][/tex]
2. Factor the quadratic expression:
Now, focus on factoring the quadratic expression [tex]\( x^2 - 7x + 10 \)[/tex]. We need to find two numbers that multiply to 10 (the constant term) and add up to -7 (the coefficient of the middle term).
The numbers that satisfy these conditions are -5 and -2. Therefore, we can write:
[tex]\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\][/tex]
3. Write the completely factored form:
Substituting the factored form of the quadratic expression back into the original expression, we get:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
So, the expression [tex]\( x^3 - 7x^2 + 10x \)[/tex] is completely factored as [tex]\( x(x - 5)(x - 2) \)[/tex].
1. Look for common factors:
Start by identifying any common factors in the terms of the expression. Here, each term contains a factor of [tex]\( x \)[/tex]. So, factor [tex]\( x \)[/tex] out of the expression:
[tex]\[
x(x^2 - 7x + 10)
\][/tex]
2. Factor the quadratic expression:
Now, focus on factoring the quadratic expression [tex]\( x^2 - 7x + 10 \)[/tex]. We need to find two numbers that multiply to 10 (the constant term) and add up to -7 (the coefficient of the middle term).
The numbers that satisfy these conditions are -5 and -2. Therefore, we can write:
[tex]\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\][/tex]
3. Write the completely factored form:
Substituting the factored form of the quadratic expression back into the original expression, we get:
[tex]\[
x(x - 5)(x - 2)
\][/tex]
So, the expression [tex]\( x^3 - 7x^2 + 10x \)[/tex] is completely factored as [tex]\( x(x - 5)(x - 2) \)[/tex].
