Problem 51 Show that $$ (a+b)^{3}=a^{3}... [FREE SOLUTION]

91影视

Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 2: Problem 51

Show that $$ (a+b)^{3}=a^{3}+b^{3} $$ if and only if $a=0$ or $b=0$ or $a=-b$.

Short Answer

Expert verified

The given equation $(a+b)^3=a^3+b^3$ holds true if and only if one of the following conditions is met: $a = 0$, $b = 0$, or $a = -b$.

Step by step solution

Expand the Left Side using the Binomial Theorem

We start by expanding $(a+b)^3$ using the binomial theorem, which states that for any non-negative integer n and any real numbers a and b: $ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k $ Applying this to our case with $n=3$, we have: \[ (a+b)^3 = \sum_{k=0}^{3} \binom{3}{k} a^{3-k} b^k \] Now, we can compute the individual terms of the sum.

Compute the Individual Terms of the Sum

We compute the individual terms of the sum as follows: $ \binom{3}{0} a^{3-0} b^0 = 1 \cdot a^3 \cdot 1 = a^3 \\ \binom{3}{1} a^{3-1} b^1 = 3 \cdot a^2 \cdot b = 3a^2b \\ \binom{3}{2} a^{3-2} b^2 = 3 \cdot a \cdot b^2 = 3ab^2 \\ \binom{3}{3} a^{3-3} b^3 = 1 \cdot 1 \cdot b^3 = b^3 $ So, we can rewrite the left side of the equation as: \[ (a+b)^3=a^3+3a^2b+3ab^2+b^3 \]

Comparing both sides of the equation and finding the conditions

Now, we have the expanded form of the equation: \[ a^3+3a^2b+3ab^2+b^3 = a^3+b^3 \] By comparing both sides, we observe that the equation holds true when the following term is equal to 0: \[ 3a^2b+3ab^2=0 \] We can factor out $3ab$ and rewrite the equation as: \[ 3ab(a+b)=0 \] For this equation to be true, one of the factors must equal 0. Therefore, we have three cases: 1) $a = 0$ 2) $b = 0$ 3) $a + b = 0$ which implies $a = -b$ So, the given equation holds true if and only if $a = 0$ or $b = 0$ or $a = -b$.

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91影视

Show that $$ (a+b)^{3}=a^{3}+b^{3} $$ if and only if \(a=0\) or \(b=0\) or \(a=-b\).

Short Answer

Step by step solution

Expand the Left Side using the Binomial Theorem

Compute the Individual Terms of the Sum

Comparing both sides of the equation and finding the conditions

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