Question
Factor completely.
3d2–10d+8
3d2–10d+8
Answers
GPT-4o mini
To factor the expression \(3d^2 - 10d + 8\), we can use the factoring by grouping method or try to find two numbers that multiply to the product of the coefficient of \(d^2\) (which is 3) and the constant term (which is 8), and add up to the coefficient of \(d\) (which is -10).
1. **Multiply the coefficient of \(d^2\) and the constant term:**
\[
3 \times 8 = 24
\]
2. **We need to find two numbers that multiply to 24 and add up to -10.** The numbers that fit this are -6 and -4, since:
\[
-6 \times -4 = 24
\]
\[
-6 + (-4) = -10
\]
3. **Rewrite the middle term (-10d) using -6d and -4d:**
\[
3d^2 - 6d - 4d + 8
\]
4. **Factor by grouping:**
- Group the first two terms and the last two terms:
\[
(3d^2 - 6d) + (-4d + 8)
\]
- Factor out the common factors from each group:
\[
3d(d - 2) - 4(d - 2)
\]
5. **Now, factor out the common binomial factor \((d - 2)\):**
\[
(3d - 4)(d - 2)
\]
Thus, the completely factored form of \(3d^2 - 10d + 8\) is:
\[
(3d - 4)(d - 2)
\]
1. **Multiply the coefficient of \(d^2\) and the constant term:**
\[
3 \times 8 = 24
\]
2. **We need to find two numbers that multiply to 24 and add up to -10.** The numbers that fit this are -6 and -4, since:
\[
-6 \times -4 = 24
\]
\[
-6 + (-4) = -10
\]
3. **Rewrite the middle term (-10d) using -6d and -4d:**
\[
3d^2 - 6d - 4d + 8
\]
4. **Factor by grouping:**
- Group the first two terms and the last two terms:
\[
(3d^2 - 6d) + (-4d + 8)
\]
- Factor out the common factors from each group:
\[
3d(d - 2) - 4(d - 2)
\]
5. **Now, factor out the common binomial factor \((d - 2)\):**
\[
(3d - 4)(d - 2)
\]
Thus, the completely factored form of \(3d^2 - 10d + 8\) is:
\[
(3d - 4)(d - 2)
\]