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Answer :
To factor the polynomial [tex]\(3x^3 - 27x^2 + 42x\)[/tex] completely, let's follow these steps:
1. Identify the Greatest Common Factor (GCF):
Look at the coefficients of each term: 3, -27, and 42. The GCF of these numbers is 3. Additionally, the variable [tex]\(x\)[/tex] is present in each term, with the smallest power being [tex]\(x\)[/tex]. Thus, the GCF of the polynomial is [tex]\(3x\)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by the GCF, [tex]\(3x\)[/tex].
[tex]\[
3x^3 \div 3x = x^2
\][/tex]
[tex]\[
-27x^2 \div 3x = -9x
\][/tex]
[tex]\[
42x \div 3x = 14
\][/tex]
After factoring out [tex]\(3x\)[/tex], the expression becomes:
[tex]\[
3x(x^2 - 9x + 14)
\][/tex]
3. Factor the Quadratic Expression:
Now, we need to factor the quadratic [tex]\(x^2 - 9x + 14\)[/tex]. Look for two numbers that multiply to 14 and add to -9. These numbers are -2 and -7.
4. Write the Factored Form:
Using these numbers, the quadratic expression can be expressed as:
[tex]\[
x^2 - 9x + 14 = (x - 2)(x - 7)
\][/tex]
5. Combine Everything:
Put the factors together to express the completely factored form of the original polynomial:
[tex]\[
3x(x - 2)(x - 7)
\][/tex]
Thus, the polynomial [tex]\(3x^3 - 27x^2 + 42x\)[/tex] factors completely to:
[tex]\[ 3x(x - 2)(x - 7) \][/tex]
1. Identify the Greatest Common Factor (GCF):
Look at the coefficients of each term: 3, -27, and 42. The GCF of these numbers is 3. Additionally, the variable [tex]\(x\)[/tex] is present in each term, with the smallest power being [tex]\(x\)[/tex]. Thus, the GCF of the polynomial is [tex]\(3x\)[/tex].
2. Factor out the GCF:
Divide each term of the polynomial by the GCF, [tex]\(3x\)[/tex].
[tex]\[
3x^3 \div 3x = x^2
\][/tex]
[tex]\[
-27x^2 \div 3x = -9x
\][/tex]
[tex]\[
42x \div 3x = 14
\][/tex]
After factoring out [tex]\(3x\)[/tex], the expression becomes:
[tex]\[
3x(x^2 - 9x + 14)
\][/tex]
3. Factor the Quadratic Expression:
Now, we need to factor the quadratic [tex]\(x^2 - 9x + 14\)[/tex]. Look for two numbers that multiply to 14 and add to -9. These numbers are -2 and -7.
4. Write the Factored Form:
Using these numbers, the quadratic expression can be expressed as:
[tex]\[
x^2 - 9x + 14 = (x - 2)(x - 7)
\][/tex]
5. Combine Everything:
Put the factors together to express the completely factored form of the original polynomial:
[tex]\[
3x(x - 2)(x - 7)
\][/tex]
Thus, the polynomial [tex]\(3x^3 - 27x^2 + 42x\)[/tex] factors completely to:
[tex]\[ 3x(x - 2)(x - 7) \][/tex]
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