Expand the binomial \((x+2)^4\) using the Binomial Theorem, and then factor the resulting polynomial.
Answer
Step 4: Factor the polynomial. Since the polynomial is already in its simplest form, we cannot factor it further. Thus, the factored form of the polynomial is still \(x^4 + 8x^3 + 24x^2 + 32x + 16\)
Step 1 ：Step 1: Expand the binomial using the Binomial Theorem. The Binomial Theorem states that \((x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k\). So, \((x+2)^4 = {4 \choose 0} x^4 + {4 \choose 1} x^3 2 + {4 \choose 2} x^2 2^2 + {4 \choose 3} x 2^3 + {4 \choose 4} 2^4\)
Step 2 ：Step 2: Simplify the coefficients. We have, \((x+2)^4 = 1\cdot x^4 + 4\cdot x^3\cdot 2 + 6\cdot x^2\cdot 4 + 4\cdot x\cdot 8 + 1\cdot 16\)
Step 3 ：Step 3: Simplify the polynomial. Therefore, \((x+2)^4 = x^4 + 8x^3 + 24x^2 + 32x + 16\)
Step 4 ：Step 4: Factor the polynomial. Since the polynomial is already in its simplest form, we cannot factor it further. Thus, the factored form of the polynomial is still \(x^4 + 8x^3 + 24x^2 + 32x + 16\)
