Answer :
To factor the expression [tex]\(3x^4 + 45x^3 + 42x^2\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor in all the terms. Here, the terms are [tex]\(3x^4\)[/tex], [tex]\(45x^3\)[/tex], and [tex]\(42x^2\)[/tex]. Each term has a factor of [tex]\(3x^2\)[/tex]. So, [tex]\(3x^2\)[/tex] is the GCF.
2. Factor out the GCF:
When we factor [tex]\(3x^2\)[/tex] out of each term, we divide each term by [tex]\(3x^2\)[/tex]:
- From [tex]\(3x^4\)[/tex], we get [tex]\(\frac{3x^4}{3x^2} = x^2\)[/tex].
- From [tex]\(45x^3\)[/tex], we get [tex]\(\frac{45x^3}{3x^2} = 15x\)[/tex].
- From [tex]\(42x^2\)[/tex], we get [tex]\(\frac{42x^2}{3x^2} = 14\)[/tex].
So, when we factor out the GCF, the expression becomes:
[tex]\[
3x^2(x^2 + 15x + 14)
\][/tex]
3. Factor the quadratic expression:
Now, we need to factor the quadratic [tex]\(x^2 + 15x + 14\)[/tex]. We are looking for two numbers that multiply to 14 (the constant term) and add up to 15 (the coefficient of the linear term). These two numbers are [tex]\(1\)[/tex] and [tex]\(14\)[/tex].
So, we can write the quadratic as:
[tex]\[
(x + 1)(x + 14)
\][/tex]
4. Combine all factors:
Now that the quadratic is factored, we can combine it with the factored-out GCF:
[tex]\[
3x^2(x + 1)(x + 14)
\][/tex]
Thus, the completely factored form of the expression [tex]\(3x^4 + 45x^3 + 42x^2\)[/tex] is [tex]\(3x^2(x + 1)(x + 14)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor in all the terms. Here, the terms are [tex]\(3x^4\)[/tex], [tex]\(45x^3\)[/tex], and [tex]\(42x^2\)[/tex]. Each term has a factor of [tex]\(3x^2\)[/tex]. So, [tex]\(3x^2\)[/tex] is the GCF.
2. Factor out the GCF:
When we factor [tex]\(3x^2\)[/tex] out of each term, we divide each term by [tex]\(3x^2\)[/tex]:
- From [tex]\(3x^4\)[/tex], we get [tex]\(\frac{3x^4}{3x^2} = x^2\)[/tex].
- From [tex]\(45x^3\)[/tex], we get [tex]\(\frac{45x^3}{3x^2} = 15x\)[/tex].
- From [tex]\(42x^2\)[/tex], we get [tex]\(\frac{42x^2}{3x^2} = 14\)[/tex].
So, when we factor out the GCF, the expression becomes:
[tex]\[
3x^2(x^2 + 15x + 14)
\][/tex]
3. Factor the quadratic expression:
Now, we need to factor the quadratic [tex]\(x^2 + 15x + 14\)[/tex]. We are looking for two numbers that multiply to 14 (the constant term) and add up to 15 (the coefficient of the linear term). These two numbers are [tex]\(1\)[/tex] and [tex]\(14\)[/tex].
So, we can write the quadratic as:
[tex]\[
(x + 1)(x + 14)
\][/tex]
4. Combine all factors:
Now that the quadratic is factored, we can combine it with the factored-out GCF:
[tex]\[
3x^2(x + 1)(x + 14)
\][/tex]
Thus, the completely factored form of the expression [tex]\(3x^4 + 45x^3 + 42x^2\)[/tex] is [tex]\(3x^2(x + 1)(x + 14)\)[/tex].
