Question

write the following expression as a simplified polynomial in standard form
$(x - 4)^2 + 5(x - 4) - 1$

Explanation:

Step1: Expand \((x - 4)^2\)

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), where \(a = x\) and \(b = 4\), we get:
\((x - 4)^2 = x^2 - 8x + 16\)

Step2: Expand \(5(x - 4)\)

Using the distributive property \(a(b - c)=ab - ac\), where \(a = 5\), \(b = x\), and \(c = 4\), we get:
\(5(x - 4)=5x - 20\)

Step3: Substitute the expanded forms into the original expression

Substitute \((x - 4)^2 = x^2 - 8x + 16\) and \(5(x - 4)=5x - 20\) into \((x - 4)^2 + 5(x - 4)-1\):
\(x^2 - 8x + 16 + 5x - 20 - 1\)

Step4: Combine like terms

Combine the \(x\)-terms: \(-8x + 5x=-3x\)
Combine the constant terms: \(16 - 20 - 1=-5\)
So the simplified polynomial is \(x^2 - 3x - 5\)

Answer:

\(x^2 - 3x - 5\)