Question
Question Write the following expression as a simplified polynomial in standard form \[ 3(x+4)^{2}+2(x+4)+4 \]
Ask by Moran Deleon. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified polynomial in standard form is \( 3x^2 + 26x + 60 \).
Solution
First, we'll expand the square term:
(x + 4)² = x² + 8x + 16.
Multiply by 3:
3(x² + 8x + 16) = 3x² + 24x + 48.
Now, expand the middle term:
2(x + 4) = 2x + 8.
Finally, add all terms together with the constant 4:
3x² + 24x + 48 + 2x + 8 + 4.
Combine like terms:
x² terms: 3x².
x terms: 24x + 2x = 26x.
Constant terms: 48 + 8 + 4 = 60.
The simplified polynomial in standard form is:
3x² + 26x + 60.
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Simplify this solution The Deep Dive
To simplify the expression \( 3(x+4)^{2}+2(x+4)+4 \), first, expand \( (x + 4)^{2} \), which gives \( x^{2} + 8x + 16 \). Now substitute this back into the expression: \[ 3(x^2 + 8x + 16) + 2(x + 4) + 4 \] Distributing the \( 3 \): \[ 3x^2 + 24x + 48 + 2x + 8 + 4 \] Now combine like terms: \[ 3x^2 + (24x + 2x) + (48 + 8 + 4) \] Which simplifies to: \[ 3x^2 + 26x + 60 \] So, the simplified polynomial in standard form is: \[ 3x^2 + 26x + 60 \]
