Pregunta
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Respuesta verificada por el tutor
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\( x^{3}+12x^{2}+36x = x(x+6)^2 \)
\( 4x^{2}-28x+49 = (2x-7)^2 \)
Solución
1. Factorize \( x^{3}+12x^{2}+36x \):
- First, factor out the common factor \( x \):
\[
x^{3}+12x^{2}+36x = x\left(x^{2}+12x+36\right)
\]
- Notice that the quadratic expression is a perfect square:
\[
x^{2}+12x+36 = \left(x+6\right)^{2}
\]
- Therefore, the complete factorization is:
\[
x\left(x+6\right)^{2}
\]
2. Factorize \( 4x^{2}-28x+49 \):
- Recognize that the quadratic is a perfect square:
\[
4x^{2}-28x+49 = \left(2x-7\right)^{2}
\]
- This can be verified by expanding:
\[
\left(2x-7\right)^{2} = 4x^{2}-28x+49
\]
The final answers are:
- \( x^{3}+12x^{2}+36x = x\left(x+6\right)^{2} \)
- \( 4x^{2}-28x+49 = \left(2x-7\right)^{2} \)
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución Mind Expander
Let's talk about the first expression! Did you know that the polynomial \( x^{3}+12 x^{2}+36 x \) can be factored by taking out the greatest common factor (GCF)? Here, the GCF is \( x \), so you can rewrite it as \( x(x^{2}+12 x+36) \). The quadratic can be further factored into \( x(x+6)^{2} \). Voilà! Now, moving on to the second expression, \( 4 x^{2}-28 x+49 \). This is a fun one because it resembles a perfect square trinomial! You can express it as \( (2x - 7)^{2} \). Perfect squares are super handy in algebra, as they make solving equations a breeze—no fancy quadratic formula needed for this one!
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