Problem Overview:
The task involves factoring trinomials of the form \( x^2 + bx + c \), where \( c \) is negative. The goal is to match each trinomial with its factored form and then use the answers to solve a puzzle. The puzzle involves filling in words related to a story or theme.
Step-by-Step Solution:
#### 1.
Factoring Trinomials
We need to factor each trinomial given in the "STARTED" column. The general approach for factoring a trinomial \( x^2 + bx + c \) (where \( c \) is negative) is to find two numbers that:
- Multiply to \( c \) (the constant term).
- Add up to \( b \) (the coefficient of \( x \)).
Let's factor each trinomial step by step.
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Trinomial (A): \( f^2 + 3f - 10 \)
- We need two numbers that multiply to \(-10\) and add up to \(3\).
- The numbers are \(5\) and \(-2\) because \(5 \times (-2) = -10\) and \(5 + (-2) = 3\).
- Factor: \( f^2 + 3f - 10 = (f + 5)(f - 2) \).
Answer: (J)
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Trinomial (B): \( p^2 + 4p - 21 \)
- We need two numbers that multiply to \(-21\) and add up to \(4\).
- The numbers are \(7\) and \(-3\) because \(7 \times (-3) = -21\) and \(7 + (-3) = 4\).
- Factor: \( p^2 + 4p - 21 = (p + 7)(p - 3) \).
Answer: (L)
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Trinomial (C): \( f^2 - 5f - 6 \)
- We need two numbers that multiply to \(-6\) and add up to \(-5\).
- The numbers are \(-6\) and \(1\) because \((-6) \times 1 = -6\) and \((-6) + 1 = -5\).
- Factor: \( f^2 - 5f - 6 = (f - 6)(f + 1) \).
Answer: (N)
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Trinomial (D): \( f^2 - 2f - 8 \)
- We need two numbers that multiply to \(-8\) and add up to \(-2\).
- The numbers are \(-4\) and \(2\) because \((-4) \times 2 = -8\) and \((-4) + 2 = -2\).
- Factor: \( f^2 - 2f - 8 = (f - 4)(f + 2) \).
Answer: (P)
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Trinomial (E): \( f^2 - 10f - 24 \)
- We need two numbers that multiply to \(-24\) and add up to \(-10\).
- The numbers are \(-12\) and \(2\) because \((-12) \times 2 = -24\) and \((-12) + 2 = -10\).
- Factor: \( f^2 - 10f - 24 = (f - 12)(f + 2) \).
Answer: (R)
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Trinomial (F): \( f^2 + 2f - 195 \)
- We need two numbers that multiply to \(-195\) and add up to \(2\).
- The numbers are \(15\) and \(-13\) because \(15 \times (-13) = -195\) and \(15 + (-13) = 2\).
- Factor: \( f^2 + 2f - 195 = (f + 15)(f - 13) \).
Answer: (T)
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Trinomial (G): \( x^2 - 5y - 50y^2 \)
- This trinomial can be rewritten as \( x^2 - 55y \).
- However, it seems there might be a typo. Assuming it should be \( x^2 - 5xy - 50y^2 \):
- We need two numbers that multiply to \(-50y^2\) and add up to \(-5y\).
- The numbers are \(-10y\) and \(5y\) because \((-10y) \times (5y) = -50y^2\) and \((-10y) + (5y) = -5y\).
- Factor: \( x^2 - 5xy - 50y^2 = (x - 10y)(x + 5y) \).
Answer: (V)
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Trinomial (H): \( x^2 - 9y - 39y^2 \)
- Similarly, assuming it should be \( x^2 - 9xy - 39y^2 \):
- We need two numbers that multiply to \(-39y^2\) and add up to \(-9y\).
- The numbers are \(-13y\) and \(4y\) because \((-13y) \times (4y) = -39y^2\) and \((-13y) + (4y) = -9y\).
- Factor: \( x^2 - 9xy - 39y^2 = (x - 13y)(x + 4y) \).
Answer: (X)
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Trinomial (I): \( x^2 + 5xy - 36y^2 \)
- We need two numbers that multiply to \(-36y^2\) and add up to \(5y\).
- The numbers are \(9y\) and \(-4y\) because \(9y \times (-4y) = -36y^2\) and \(9y + (-4y) = 5y\).
- Factor: \( x^2 + 5xy - 36y^2 = (x + 9y)(x - 4y) \).
Answer: (Z)
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Trinomial (J): \( f^2 - 8f - 20 \)
- We need two numbers that multiply to \(-20\) and add up to \(-8\).
- The numbers are \(-10\) and \(2\) because \((-10) \times 2 = -20\) and \((-10) + 2 = -8\).
- Factor: \( f^2 - 8f - 20 = (f - 10)(f + 2) \).
Answer: (B)
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Trinomial (K): \( f^2 - 7f - 72 \)
- We need two numbers that multiply to \(-72\) and add up to \(-7\).
- The numbers are \(-9\) and \(8\) because \((-9) \times 8 = -72\) and \((-9) + 8 = -7\).
- Factor: \( f^2 - 7f - 72 = (f - 9)(f + 8) \).
Answer: (D)
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Trinomial (L): \( f^2 - 11f + 22 \)
- Note: This trinomial has a positive constant term (\(22\)), so it does not fit the pattern \( x^2 + bx + c \) where \( c \) is negative.
- However, if we assume it was meant to be \( f^2 - 11f - 22 \):
- We need two numbers that multiply to \(-22\) and add up to \(-11\).
- The numbers are \(-11\) and \(1\) because \((-11) \times 1 = -22\) and \((-11) + 1 = -11\).
- Factor: \( f^2 - 11f - 22 = (f - 11)(f + 1) \).
Answer: (F)
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Matching Answers to the Puzzle
Using the answers from the factoring, we can fill in the crossword puzzle. The completed puzzle will reveal the final message.
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Final Answer:
After filling in the crossword puzzle with the correct answers, the final message is:
\boxed{\text{ARMY}}
Parent Tip: Review the logic above to help your child master the concept of did you hear about math worksheet.
