Question

factor the trinomial. if the trinomial cannot be factored, write prime. (ex. 2 on notes)
5.) $x^2 + 3x + 2$ 6.) $y^2 + 4y + 4$ 7.) $z^2 + 3z + 7$
8.) $x^2 + 8x + 15$ 9.) $m^2 + 13m + 36$ 10.) $n^2 + 20n + 100$
11.) $x^2 - 6x + 5$ 12.) $y^2 - 7y + 12$ 13.) $z^2 - 13z + 40$
14.) $a^2 - 16a + 63$ 15.) $y^2 - 6y + 10$ 16.) $b^2 - 30b + 125$
17.) $x^2 + 13x - 90$ 18.) $m^2 + 4m - 45$ 19.) $a^2 + 16a - 63$
20.) $n^2 + 10n - 200$ 21.) $x^2 + 22x - 23$ 22.) $a^2 + 4a - 32$
23.) $b^2 - b - 20$ 24.) $m^2 - 14m - 22$ 25.) $x^2 - x - 72$

Answer:

To solve these trinomial factoring problems, we'll use the method of finding two numbers that multiply to the constant term and add up to the coefficient of the middle term (for trinomials of the form \(x^2 + bx + c\)). Let's solve a few examples:

Problem 5: \(x^2 + 3x + 2\)

Step 1: Find two numbers that multiply to \(2\) and add to \(3\).

The numbers are \(1\) and \(2\) (since \(1 \times 2 = 2\) and \(1 + 2 = 3\)).

Step 2: Factor the trinomial.

Using these numbers, we can write the trinomial as \((x + 1)(x + 2)\).

Problem 6: \(y^2 + 4y + 4\)

Step 1: Find two numbers that multiply to \(4\) and add to \(4\).

The numbers are \(2\) and \(2\) (since \(2 \times 2 = 4\) and \(2 + 2 = 4\)).

Step 2: Factor the trinomial.

This is a perfect square trinomial, so it factors as \((y + 2)^2\).

Problem 7: \(z^2 + 3z + 7\)

Step 1: Check if two numbers multiply to \(7\) and add to \(3\).

The factors of \(7\) are \(1\) and \(7\) (since \(7\) is prime). \(1 + 7 = 8
eq 3\), so no such numbers exist.

Step 2: Determine if it's prime.

Since we can't find two numbers that satisfy the conditions, the trinomial is prime.

Problem 8: \(x^2 + 8x + 15\)

Step 1: Find two numbers that multiply to \(15\) and add to \(8\).

The numbers are \(3\) and \(5\) (since \(3 \times 5 = 15\) and \(3 + 5 = 8\)).

Step 2: Factor the trinomial.

Using these numbers, we get \((x + 3)(x + 5)\).

Problem 9: \(m^2 + 13m + 36\)

Step 1: Find two numbers that multiply to \(36\) and add to \(13\).

The numbers are \(4\) and \(9\) (since \(4 \times 9 = 36\) and \(4 + 9 = 13\)).

Step 2: Factor the trinomial.

This factors as \((m + 4)(m + 9)\).

Problem 10: \(n^2 + 20n + 100\)

Step 1: Find two numbers that multiply to \(100\) and add to \(20\).

The numbers are \(10\) and \(10\) (since \(10 \times 10 = 100\) and \(10 + 10 = 20\)).

Step 2: Factor the trinomial.

This is a perfect square trinomial, so it factors as \((n + 10)^2\).

Problem 11: \(x^2 - 6x + 5\)

Step 1: Find two numbers that multiply to \(5\) and add to \(-6\) (note the signs: both numbers will be negative since their product is positive and their sum is negative).

The numbers are \(-1\) and \(-5\) (since \((-1) \times (-5) = 5\) and \(-1 + (-5) = -6\)).

Step 2: Factor the trinomial.

Using these numbers, we get \((x - 1)(x - 5)\).

Problem 12: \(y^2 - 7y + 12\)

Step 1: Find two numbers that multiply to \(12\) and add to \(-7\) (both numbers negative).

The numbers are \(-3\) and \(-4\) (since \((-3) \times (-4) = 12\) and \(-3 + (-4) = -7\)).

Step 2: Factor the trinomial.

This factors as \((y - 3)(y - 4)\).

Problem 13: \(z^2 - 13z + 40\)

Step 1: Find two numbers that multiply to \(40\) and add to \(-13\) (both numbers negative).

The numbers are \(-5\) and \(-8\) (since \((-5) \times (-8) = 40\) and \(-5 + (-8) = -13\)).

Step 2: Factor the trinomial.

This factors as \((z - 5)(z - 8)\).

Problem 14: \(a^2 - 16a + 63\)

Step 1: Find two numbers that multiply to \(63\) and add to \(-16\) (both numbers negative).

The numbers are \(-7\) and \(-9\) (since \((-7) \times (-9) = 63\) and \(-7 + (-9) = -16\)).

Step 2: Factor the trinomial.

This factors as \((a - 7)(a - 9)\).

Problem 15: \(y^2 - 6y + 10\)

Step 1: Check if two numbers multiply to \(10\) and add to \(-6\).

The factors of \(10\) are \(1\) and \(10\), \(2\) and \(5\). None of these pairs add up to \(-6\) (even considering negative factors: \(-1 + (-10) = -11\), \(-2 + (-5) = -7\)).

Step 2: Determine if it's prime.

Since we can't find two numbers that satisfy the conditions, the trinomial is prime.

Problem 16: \(b^2 - 30b + 125\)

Step 1: Find two numbers that multiply to \(125\) and add to \(-30\) (both numbers negative).

The factors of \(125\) are \(1\) and \(125\), \(5\) and \(25\). Checking negative pairs: \(-5 + (-25) = -30\) (since \((-5) \times (-25) = 125\)).

Step 2: Factor the trinomial.

This factors as \((b - 5)(b - 25)\).

Problem 17: \(x^2 + 13x - 90\)

Step 1: Find two numbers that multiply to \(-90\) and add to \(13\) (one positive, one negative; the positive number has a larger absolute value).

The numbers are \(18\) and \(-5\) (since \(18 \times (-5) = -90\) and \(18 + (-5) = 13\)).

Step 2: Factor the trinomial.

This factors as \((x + 18)(x - 5)\).

Problem 18: \(m^2 + 4m - 45\)

Step 1: Find two numbers that multiply to \(-45\) and add to \(4\) (one positive, one negative; the positive number has a larger absolute value).

The numbers are \(9\) and \(-5\) (since \(9 \times (-5) = -45\) and \(9 + (-5) = 4\)).

Step 2: Factor the trinomial.

This factors as \((m + 9)(m - 5)\).

Problem 19: \(a^2 + 16a - 63\)

Step 1: Check if two numbers multiply to \(-63\) and add to \(16\).

The factors of \(63\) are \(1\) and \(63\), \(3\) and \(21\), \(7\) and \(9\). Checking pairs with opposite signs: \(21\) and \(-3\) (since \(21 \times (-3) = -63\) and \(21 + (-3) = 18
eq 16\)); \(9\) and \(-7\) (since \(9 \times (-7) = -63\) and \(9 + (-7) = 2
eq 16\)); \(63\) and \(-1\) (since \(63 \times (-1) = -63\) and \(63 + (-1) = 62
eq 16\)). No such numbers exist.

Step 2: Determine if it's prime.

Since we can't find two numbers that satisfy the conditions, the trinomial is prime.

Problem 20: \(n^2 + 10n - 200\)

Step 1: Find two numbers that multiply to \(-200\) and add to \(10\) (one positive, one negative; the positive number has a larger absolute value).

The numbers are \(20\) and \(-10\) (since \(20 \times (-10) = -200\) and \(20 + (-10) = 10\)).

Step 2: Factor the trinomial.

This factors as \((n + 20)(n - 10)\).

Problem 21: \(x^2 + 22x - 23\)

Step 1: Find two numbers that multiply to \(-23\) and add to \(22\) (one positive, one negative; the positive number has a larger absolute value).

The numbers are \(23\) and \(-1\) (since \(23 \times (-1) = -23\) and \(23 + (-1) = 22\)).

Step 2: Factor the trinomial.

This factors as \((x + 23)(x - 1)\).

Problem 22: \(a^2 + 4a - 32\)

Step 1: Find two numbers that multiply to \(-32\) and add to \(4\) (one positive, one negative; the positive number has a larger absolute value).

The numbers are \(8\) and \(-4\) (since \(8 \times (-4) = -32\) and \(8 + (-4) = 4\)).

Step 2: Factor the trinomial.

This factors as \((a + 8)(a - 4)\).

Problem 23: \(b^2 - b - 20\)

Step 1: Find two numbers that multiply to \(-20\) and add to \(-1\) (one positive, one negative; the negative number has a larger absolute value).

The numbers are \(4\) and \(-5\) (since \(4 \times (-5) = -20\) and \(4 + (-5) = -1\)).

Step 2: Factor the trinomial.

This factors as \((b + 4)(b - 5)\).

Problem 24: \(m^2 - 14m - 22\)

Step 1: Check if two numbers multiply to \(-22\) and add to \(-14\).

The factors of \(22\) are \(1\) and \(22\), \(2\) and \(11\). Checking pairs with opposite signs: \( -11\) and \(2\) (since \(-11 \times 2 = -22\) and \(-11 + 2 = -9
eq -14\)); \( -22\) and \(1\) (since \(-22 \times 1 = -22\) and \(-22 + 1 = -21
eq -14\)). No such numbers exist.

Step 2: Determine if it's prime.

Since we can't find two numbers that satisfy the conditions, the trinomial is prime.

Problem 25: \(x^2 - x - 72\)

Step 1: Find two numbers that multiply to \(-72\) and add to \(-1\) (one positive, one negative; the negative number has a larger absolute value).

The numbers are \(8\) and \(-9\) (since \(8 \times (-9) = -72\) and \(8 + (-9) = -1\)).

Step 2: Factor the trinomial.

This factors as \((x + 8)(x - 9)\).

Final Answers:

1. \(\boldsymbol{(x + 1)(x + 2)}\)
2. \(\boldsymbol{(y + 2)^2}\)
3. \(\boldsymbol{\text{prime}}\)
4. \(\boldsymbol{(x + 3)(x + 5)}\)
5. \(\boldsymbol{(m + 4)(m + 9)}\)
6. \(\boldsymbol{(n + 10)^2}\)
7. \(\boldsymbol{(x - 1)(x - 5)}\)
8. \(\boldsymbol{(y - 3)(y - 4)}\)
9. \(\boldsymbol{(z - 5)(z - 8)}\)
10. \(\boldsymbol{(a - 7)(a - 9)}\)
11. \(\boldsymbol{\text{prime}}\)
12. \(\boldsymbol{(b - 5)(b - 25)}\)
13. \(\boldsymbol{(x + 18)(x - 5)}\)
14. \(\boldsymbol{(m + 9)(m - 5)}\)
15. \(\boldsymbol{\text{prime}}\)
16. \(\boldsymbol{(n + 20)(n - 10)}\)
17. \(\boldsymbol{(x + 23)(x - 1)}\)
18. \(\boldsymbol{(a + 8)(a - 4)}\)
19. \(\boldsymbol{(b + 4)(b - 5)}\)
20. \(\boldsymbol{\text{prime}}\)
21. \(\boldsymbol{(x + 8)(x - 9)}\)