To factor the trinomial [tex]\(6x^2 - 17x + 3\)[/tex], we will use the method of factoring by grouping. Here are the steps in detail:
1. Identify the coefficients:
The trinomial is [tex]\(6x^2 - 17x + 3\)[/tex], where the coefficient of [tex]\(x^2\)[/tex] (a) is 6, the coefficient of [tex]\(x\)[/tex] (b) is -17, and the constant term (c) is 3.
2. Multiply [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
Multiply the coefficients of [tex]\(x^2\)[/tex] and the constant term, [tex]\(6 \times 3 = 18\)[/tex].
3. Find two numbers that multiply to [tex]\(ac\)[/tex] and add to [tex]\(b\)[/tex]:
We need to find two numbers that multiply to [tex]\(18\)[/tex] and add to [tex]\(-17\)[/tex]. These numbers are [tex]\(-1\)[/tex] and [tex]\(-18\)[/tex] because:
[tex]\[
(-1) \times (-18) = 18 \quad \text{and} \quad (-1) + (-18) = -19.
\][/tex]
4. Rewrite the middle term using these two numbers:
Rewrite [tex]\(-17x\)[/tex] as [tex]\(-1x - 18x\)[/tex]:
[tex]\[
6x^2 - 1x - 18x + 3
\][/tex]
5. Factor by grouping:
Group the terms to factor by grouping:
[tex]\[
(6x^2 - 1x) - (18x - 3)
\][/tex]
Factor out the greatest common factor (GCF) from each group:
[tex]\[
x(6x - 1) - 3(6x - 1)
\][/tex]
6. Factor out the common binomial factor:
Both groups contain [tex]\((6x - 1)\)[/tex], so factor this out:
[tex]\[
(6x - 1)(x - 3)
\][/tex]
So, the factored form of the trinomial [tex]\(6x^2 - 17x + 3\)[/tex] is:
[tex]\[
(6x - 1)(x - 3)
\][/tex]
