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Common Core: High School - Algebra : Using the Binomial Theorem for Expansion: CCSS.Math.Content.HSA-APR.C.5

Study concepts, example questions & explanations for Common Core: High School - Algebra

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All Common Core: High School - Algebra Resources

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Example Questions

Example Question #11 : Using The Binomial Theorem For Expansion: Ccss.Math.Content.Hsa Apr.C.5

Use Pascal's Triangle to Expand

$\left(x + y\right)^{2}$

Possible Answers:

$x^{2} + 2 x y + y^{2}$

Not Possible

$x^{2} + 2 x y + y$

$x^{2} + 2 x y + y^{2}$

$x^{2} + y^{2}$

Correct answer:

$x^{2} + 2 x y + y^{2}$

Explanation:

In order to do this, we need to recall the formula for Pascal's Triangle.

$\sum_{i=0}^{n} x^{- i + n} y^{i} {\binom{n}{i}}$

Since the exponent in the question is 2 we can replace $\uptext{n}$ , with 2 .

Now our equation looks like

$\sum_{i=0}^{2} x^{- i + 2} y^{i} {\binom{2}{i}}$

Now we compute the sum, term by term.

Term 1 : $\binom{ 2 }{ 0 } x ^{\left( 2 - 0 \right)} y ^{ 0 }= x^{2}$

Term 2 : $\binom{ 2 }{ 1 } x ^{\left( 2 - 1 \right)} y ^{ 1 }= 2 x y$

Term 3 : $\binom{ 2 }{ 2 } x ^{\left( 2 - 2 \right)} y ^{ 2 }= y^{2}$

Now we combine the expressions and we get

$x^{2} + 2 x y + y^{2}$

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Example Question #12 : Using The Binomial Theorem For Expansion: Ccss.Math.Content.Hsa Apr.C.5

What is the coefficient of $x^{12} y^{2}$ in the expansion of $\left(x + y\right)^{14}$ ?

Possible Answers:

There is no coefficient

Correct answer:

Explanation:

In order to do this, we need to recall the formula for Pascal's Triangle.

$\sum_{i=0}^{n} x^{- i + n} y^{i} {\binom{n}{i}}$

The part in the expression that we care about is the combination.

We simply do this by looking at the exponent of $\uptext{y}$ , and the exponent of the original equation.

In this case n = 14 and i = 2

Now we compute the following

${\binom{n}{i}} = \binom{ 14 }{ 2 } = 91$

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All Common Core: High School - Algebra Resources

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Common Core: High School - Algebra : Using the Binomial Theorem for Expansion: CCSS.Math.Content.HSA-APR.C.5

Study concepts, example questions & explanations for Common Core: High School - Algebra

All Common Core: High School - Algebra Resources

Example Questions

Example Question #11 : Using The Binomial Theorem For Expansion: Ccss.Math.Content.Hsa Apr.C.5

Example Question #12 : Using The Binomial Theorem For Expansion: Ccss.Math.Content.Hsa Apr.C.5

All Common Core: High School - Algebra Resources

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