Exercise 59 Page 582

a To expand the given binomial, we should recall the Binomial Theorem. It states that for every positive integer

n,

we can expand the expression

(a + b)^{n}

by using the numbers in the

n^{th}

row of Pascal's Triangle.

(a + b)^{n} = P_{0} a^{n} b^{0} + P_{1} a^{n - 1} b^{1} + ⋮ + P_{n - 1} a^{1} b^{n - 1} + P_{n} a^{0} b^{n}

In the above formula,

P_{0},

P_{1},

\dots,

P_{n}

are the numbers in the

n^{th}

row of Pascal's Triangle.

Row Pascal ’ s Triangle 01234511151413101261310141511

Note that each number greater than

1

found in the triangle is the sum of the two numbers diagonally above it. Now consider the given binomial.

(a + b)^{4}

We can substitute the first term for

a

and the second term for

b

using the Binomial Theorem equation and the coefficients from Pascal's Triangle.

$(a + b)^{n} = P_{0} a^{n} b^{0} + P_{1} a^{n - 1} b^{1} + \dots + P_{n - 1} a^{1} b^{n - 1} + P_{n} a^{0} b^{n}$
$(a + b)^{4} = 1 a^{4} b^{0} + 4 a^{3} b^{1} + 6 a^{2} b^{2} + 4 a^{1} b^{3} + 1 a^{0} b^{4}$

Finally, let's simplify the expression.

1 a^{4} b^{0} + 4 a^{3} b^{1} + 6 a^{2} b^{2} + 4 a^{1} b^{3} + 1 a^{0} b^{4}

▼

Simplify

ExponentZero

$a^{0} = 1$

1 a^{4} (1) + 4 a^{3} b^{1} + 6 a^{2} b^{2} + 4 a^{1} b^{3} + 1 (1) b^{4}

ExponentOne

$a^{1} = a$

1 a^{4} (1) + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 (1) b^{4}

MultByOne

$a \cdot 1 = a$

a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}

b We want to expand the following expression.

(3 m - 2)^{4}

Notice that we can rewrite it so that it resembles the expression from Part A.

(3 m - 2)^{4} \Leftrightarrow ((3 m) + (- 2))^{4}

Now, if we denote

a = 3 m

and

b = - 2,

the above equation would match the one from Part A. Therefore, We can use the expanded expression from Part A by substituting

a = 3 m

and

b = - 2

and simplifying the result.

a^{4} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + b^{4}

SubstituteII

$a = 3 m$ , $b = - 2$

(3 m)^{4} + 4 (3 m)^{3} (- 2) + 6 (3 m)^{2} (- 2)^{2} + 4 (3 m) (- 2)^{3} + (- 2)^{4}

▼

Simplify

PowPow

$(a^{m})^{n} = a^{m \cdot n}$

3^{4} m^{4} + 3 (3^{3}) m^{3} (- 2) + 6 (3^{2}) m^{2} (- 2)^{2} + 4 (3 m) (- 2)^{3} + (- 2)^{4}

CalcPow

Calculate power

81 m^{4} + 3 (27) m^{3} (- 2) + 6 (9) m^{2} (4) + 4 (3 m) (- 8) + 16

Multiply

81 m^{4} - 162 m^{3} + 216 m^{2} - 72 m + 16

Subchapter links

11.2.1 Creating a Three-Dimensional Model p.582-583

11.2.2 Graphing Equations in Three Dimensions p.586-587

11.2.3 Solving Systems of Three Equations with Three Variables p.590-592

11.2.4 Using Systems of Three Equations for Curve Fitting p.598-600

100

101

102

103

104

105

106

107

108

109

110

111

112

113