Answer :
To factor the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] completely, let's work through it step by step.
1. Look for Common Factors:
The first step in factoring is to check for any common factors in all terms. We see that each term has an [tex]\( x \)[/tex], so we can factor out the greatest common factor (GCF), which is [tex]\( x \)[/tex].
[tex]\[
x^3 - 7x^2 + 10x = x(x^2 - 7x + 10)
\][/tex]
2. Factor the Quadratic:
Now, we need to factor the quadratic expression inside the parentheses: [tex]\( x^2 - 7x + 10 \)[/tex].
To factor this quadratic, we look for two numbers that multiply to 10 (the constant term) and add to -7 (the coefficient of the middle term, [tex]\( x \)[/tex]).
The numbers -2 and -5 satisfy these conditions because:
[tex]\[
(-2) \times (-5) = 10
\][/tex]
[tex]\[
(-2) + (-5) = -7
\][/tex]
Therefore, we can express the quadratic as:
[tex]\[
x^2 - 7x + 10 = (x - 2)(x - 5)
\][/tex]
3. Write the Complete Factored Form:
Substituting back into the expression we factored out initially, we have:
[tex]\[
x(x^2 - 7x + 10) = x(x - 2)(x - 5)
\][/tex]
So the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] is completely factored as [tex]\( x(x - 2)(x - 5) \)[/tex].
1. Look for Common Factors:
The first step in factoring is to check for any common factors in all terms. We see that each term has an [tex]\( x \)[/tex], so we can factor out the greatest common factor (GCF), which is [tex]\( x \)[/tex].
[tex]\[
x^3 - 7x^2 + 10x = x(x^2 - 7x + 10)
\][/tex]
2. Factor the Quadratic:
Now, we need to factor the quadratic expression inside the parentheses: [tex]\( x^2 - 7x + 10 \)[/tex].
To factor this quadratic, we look for two numbers that multiply to 10 (the constant term) and add to -7 (the coefficient of the middle term, [tex]\( x \)[/tex]).
The numbers -2 and -5 satisfy these conditions because:
[tex]\[
(-2) \times (-5) = 10
\][/tex]
[tex]\[
(-2) + (-5) = -7
\][/tex]
Therefore, we can express the quadratic as:
[tex]\[
x^2 - 7x + 10 = (x - 2)(x - 5)
\][/tex]
3. Write the Complete Factored Form:
Substituting back into the expression we factored out initially, we have:
[tex]\[
x(x^2 - 7x + 10) = x(x - 2)(x - 5)
\][/tex]
So the polynomial [tex]\( x^3 - 7x^2 + 10x \)[/tex] is completely factored as [tex]\( x(x - 2)(x - 5) \)[/tex].
