Question

factor the trinomial. if the trinomial cannot be factored, write prime. (ex. 2 on notes)
7.) 2x² + 7x + 3
8.) 3y² + 2y + 4
9.) 3x² + 4x + 1

Explanation:

7. Factoring \(2x^2 + 7x + 3\)

Step 1: Identify \(a\), \(b\), \(c\)

For \(ax^2 + bx + c\), here \(a = 2\), \(b = 7\), \(c = 3\). We need two numbers that multiply to \(a\times c=2\times3 = 6\) and add to \(b = 7\). The numbers are \(6\) and \(1\) (since \(6\times1 = 6\) and \(6 + 1=7\)).

Step 2: Split the middle term

Rewrite \(7x\) as \(6x + x\): \(2x^2+6x + x + 3\).

Step 3: Group and factor

Group the first two and last two terms: \((2x^2 + 6x)+(x + 3)\). Factor out \(2x\) from the first group and \(1\) from the second: \(2x(x + 3)+1(x + 3)\). Now factor out \((x + 3)\): \((2x + 1)(x + 3)\).

8. Factoring \(3y^2 + 2y + 4\)

Step 1: Identify \(a\), \(b\), \(c\)

Here \(a = 3\), \(b = 2\), \(c = 4\). Calculate \(a\times c=3\times4 = 12\). We need two numbers that multiply to \(12\) and add to \(2\). The factors of \(12\) are \(1\) & \(12\), \(2\) & \(6\), \(3\) & \(4\). None of these pairs add up to \(2\), so the trinomial is prime.

9. Factoring \(3x^2 + 4x + 1\)

Step 1: Identify \(a\), \(b\), \(c\)

For \(ax^2+bx + c\), \(a = 3\), \(b = 4\), \(c = 1\). \(a\times c=3\times1 = 3\), and we need two numbers that multiply to \(3\) and add to \(4\). The numbers are \(3\) and \(1\) (since \(3\times1 = 3\) and \(3 + 1 = 4\)).

Step 2: Split the middle term

Rewrite \(4x\) as \(3x+x\): \(3x^2+3x + x + 1\).

Step 3: Group and factor

Group: \((3x^2 + 3x)+(x + 1)\). Factor out \(3x\) from the first group and \(1\) from the second: \(3x(x + 1)+1(x + 1)\). Factor out \((x + 1)\): \((3x + 1)(x + 1)\).

7. Answer: \((2x + 1)(x + 3)\)

8. Answer: prime

9. Answer: \((3x + 1)(x + 1)\)

Answer:

Step 1: Identify \(a\), \(b\), \(c\)

For \(ax^2+bx + c\), \(a = 3\), \(b = 4\), \(c = 1\). \(a\times c=3\times1 = 3\), and we need two numbers that multiply to \(3\) and add to \(4\). The numbers are \(3\) and \(1\) (since \(3\times1 = 3\) and \(3 + 1 = 4\)).

Step 2: Split the middle term

Rewrite \(4x\) as \(3x+x\): \(3x^2+3x + x + 1\).

Step 3: Group and factor

Group: \((3x^2 + 3x)+(x + 1)\). Factor out \(3x\) from the first group and \(1\) from the second: \(3x(x + 1)+1(x + 1)\). Factor out \((x + 1)\): \((3x + 1)(x + 1)\).

7. Answer: \((2x + 1)(x + 3)\)

8. Answer: prime

9. Answer: \((3x + 1)(x + 1)\)