Certainly! Let's factor the quadratic polynomial [tex]\(2x^2 - 17x + 30\)[/tex].
1. Identify the coefficients: The given quadratic polynomial is [tex]\(2x^2 - 17x + 30\)[/tex]. Here, the coefficients are:
- [tex]\(a = 2\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = -17\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = 30\)[/tex] (constant term)
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] and add up to [tex]\(b\)[/tex]:
- We need two numbers that multiply to [tex]\(2 \cdot 30 = 60\)[/tex] and add up to [tex]\(-17\)[/tex].
- These two numbers are [tex]\(-20\)[/tex] and [tex]\(3\)[/tex] because [tex]\((-20) \cdot 3 = -60\)[/tex] and [tex]\((-20) + 3 = -17\)[/tex].
3. Rewrite the middle term using the two numbers found:
- We can rewrite [tex]\(-17x\)[/tex] as [tex]\(-20x + 3x\)[/tex].
- The polynomial becomes:
[tex]\[
2x^2 - 20x + 3x + 30
\][/tex]
4. Group the terms in pairs:
- Grouping terms gives us:
[tex]\[
(2x^2 - 20x) + (3x + 30)
\][/tex]
5. Factor out the greatest common factor (GCF) from each pair:
- From the first group [tex]\(2x^2 - 20x\)[/tex], we can factor out [tex]\(2x\)[/tex]:
[tex]\[
2x(x - 10)
\][/tex]
- From the second group [tex]\(3x + 30\)[/tex], we can factor out [tex]\(3\)[/tex]:
[tex]\[
3(x + 10)
\][/tex]
6. Combine the factored expressions:
- Now we have:
[tex]\[
2x(x - 10) + 3(x - 10)
\][/tex]
- We see that [tex]\((x - 10)\)[/tex] is a common factor:
[tex]\[
(x - 10)(2x + 3)
\][/tex]
However, upon verification, it seems there was an error in identifying the two numbers. So let's correct this approach.
Instead, correctly:
1. Given polynomial is [tex]\(2x^2 - 17x + 30\)[/tex].
2. The correct factored form should yield:
[tex]\[
(x - 6)(2x - 5)
\][/tex]
Thus, the factored form of the polynomial [tex]\(2x^2 - 17x + 30\)[/tex] is:
[tex]\[
(x - 6)(2*x - 5)
\][/tex]
This is the correct factorization of the given polynomial.
