Pregunta
Simplify the expression \( (x-4)\left(x^{2}+5 x+3\right) \) by multiplying the two polynomials. Answer:
Ask by Olson John. in the United States
Jan 25,2025
Solución de tutoría real
Respuesta verificada por el tutor
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\( x^{3} + x^{2} - 17x - 12 \)
Solución
To simplify the expression \( (x - 4)(x^{2} + 5x + 3) \), follow these steps:
1. **Distribute \( x \) across the second polynomial:**
\[
x \cdot x^{2} = x^{3}
\]
\[
x \cdot 5x = 5x^{2}
\]
\[
x \cdot 3 = 3x
\]
So,
\[
x(x^{2} + 5x + 3) = x^{3} + 5x^{2} + 3x
\]
2. **Distribute \( -4 \) across the second polynomial:**
\[
-4 \cdot x^{2} = -4x^{2}
\]
\[
-4 \cdot 5x = -20x
\]
\[
-4 \cdot 3 = -12
\]
So,
\[
-4(x^{2} + 5x + 3) = -4x^{2} - 20x - 12
\]
3. **Combine the results from steps 1 and 2:**
\[
x^{3} + 5x^{2} + 3x - 4x^{2} - 20x - 12
\]
4. **Combine like terms:**
\[
x^{3} + (5x^{2} - 4x^{2}) + (3x - 20x) - 12 = x^{3} + x^{2} - 17x - 12
\]
**Final Simplified Expression:**
\[
x^{3} + x^{2} - 17x - 12
\]
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Simplifique esta solución Mind Expander
To simplify the expression \( (x-4)(x^{2}+5x+3) \), we will apply the distributive property (also known as the FOIL method for binomials). 1. Multiply \( x \) by each term in the second polynomial: - \( x \cdot x^2 = x^3 \) - \( x \cdot 5x = 5x^2 \) - \( x \cdot 3 = 3x \) 2. Now multiply \( -4 \) by each term in the second polynomial: - \( -4 \cdot x^2 = -4x^2 \) - \( -4 \cdot 5x = -20x \) - \( -4 \cdot 3 = -12 \) 3. Now, combine all these results: \[ x^3 + 5x^2 + 3x - 4x^2 - 20x - 12 \] 4. Combine like terms: - \( 5x^2 - 4x^2 = 1x^2 \) - \( 3x - 20x = -17x \) Thus, the simplified expression is: \[ x^3 + x^2 - 17x - 12 \]
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