Question
When simplified, which expressions produce the trinomial 4x2−20x−24 ? Select the TWO that apply.(2 points) Responses 4(x2−5x−6) 4 times open paren x squared minus 5 x minus 6 close paren −4(x2+5x+6) negative 4 times open paren x squared plus 5 x plus 6 close paren 4(x2−20x−24) 4 times open paren x squared minus 20 x minus 24 close paren 4(x−6)(x+1) 4 open paren x minus 6 close paren open paren x plus 1 close paren
Answers
Answered by
GPT-4o mini
To determine which of the given expressions simplify to the trinomial \(4x^2 - 20x - 24\), we need to evaluate each option one by one.
1. **Option 1: \(4(x^2 - 5x - 6)\)**
Distributing the 4:
\(4 \cdot (x^2) - 4 \cdot (5x) - 4 \cdot (6) = 4x^2 - 20x - 24\)
This matches the original trinomial.
2. **Option 2: \(-4(x^2 + 5x + 6)\)**
Distributing the -4:
\(-4 \cdot (x^2) - 4 \cdot (5x) - 4 \cdot (6) = -4x^2 - 20x - 24\)
This does not match the original trinomial.
3. **Option 3: \(4(x^2 - 20x - 24)\)**
Distributing the 4:
\(4 \cdot (x^2) - 4 \cdot (20x) - 4 \cdot (24) = 4x^2 - 80x - 96\)
This does not match the original trinomial.
4. **Option 4: \(4(x - 6)(x + 1)\)**
First, we need to expand \( (x - 6)(x + 1) \):
\(x^2 + x - 6x - 6 = x^2 - 5x - 6\)
Now distributing the 4:
\(4 \cdot (x^2 - 5x - 6) = 4x^2 - 20x - 24\)
This matches the original trinomial.
Based on the evaluations above, the expressions that produce the trinomial \(4x^2 - 20x - 24\) are:
1. **\(4(x^2 - 5x - 6)\)**
2. **\(4(x - 6)(x + 1)\)**
Therefore, the two correct answers are the first and the fourth options.
1. **Option 1: \(4(x^2 - 5x - 6)\)**
Distributing the 4:
\(4 \cdot (x^2) - 4 \cdot (5x) - 4 \cdot (6) = 4x^2 - 20x - 24\)
This matches the original trinomial.
2. **Option 2: \(-4(x^2 + 5x + 6)\)**
Distributing the -4:
\(-4 \cdot (x^2) - 4 \cdot (5x) - 4 \cdot (6) = -4x^2 - 20x - 24\)
This does not match the original trinomial.
3. **Option 3: \(4(x^2 - 20x - 24)\)**
Distributing the 4:
\(4 \cdot (x^2) - 4 \cdot (20x) - 4 \cdot (24) = 4x^2 - 80x - 96\)
This does not match the original trinomial.
4. **Option 4: \(4(x - 6)(x + 1)\)**
First, we need to expand \( (x - 6)(x + 1) \):
\(x^2 + x - 6x - 6 = x^2 - 5x - 6\)
Now distributing the 4:
\(4 \cdot (x^2 - 5x - 6) = 4x^2 - 20x - 24\)
This matches the original trinomial.
Based on the evaluations above, the expressions that produce the trinomial \(4x^2 - 20x - 24\) are:
1. **\(4(x^2 - 5x - 6)\)**
2. **\(4(x - 6)(x + 1)\)**
Therefore, the two correct answers are the first and the fourth options.