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McGraw Hill Integrated II, 2012 View details

8. Differences of Squares

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Exercise 5 Page 60

The formula to factor the difference of two squares is a^2-b^2=(a+b)(a-b).

(u^2+9)(u+3)(u-3)

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Look closely at the expression u^4-81. It can be expressed as the difference of two perfect squares.

u^4-81

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Write as a power

(u^2)^2-9^2

Recall the formula to factor a difference of squares.

a^2- b^2 ⇔ ( a+ b)( a- b) We can apply this formula to our expression. ( u^2)^2- 9^2 ⇔ ( u^2+ 9)( u^2- 9) Now, let's take a look at the expression u^2-9. Once again, we can express it as a difference of two perfect squares. Let's do it!

u^2-9

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Write as a power

u^2-3^2

Now, we can apply the above formula to factor our expression completely. (u^2+9)( u^2- 3^2) ⇕ (u^2+9)( u+ 3)( u- 3)

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We can apply the Distributive Property and compare the result with the given expression.

(u^2+9)(u + 3) (u - 3)

Distr

Distribute u^2+9

(u(u^2+9) + 3(u^2+9)) (u - 3)

▼

Distribute u & 3

Distr

Distribute u

(u^3+9u + 3(u^2+9))(u - 3)

Distr

Distribute 3

(u^3+9u+3u^2+27)(u-3)

Distr

Distribute (u-3)

u^3(u-3) + 9u(u-3) + 3u^2(u-3) + 27(u-3)

▼

Simplify

Distr

Distribute u^3

u^4-3u^3 + 9u(u-3) + 3u^2(u-3) + 27(u-3)

Distr

Distribute 9u

u^4-3u^3 + 9u^2-27u + 3u^2(u-3) + 27(u-3)

Distr

Distribute 3u^2

u^4-3u^3 + 9u^2-27u + 3u^3-9u^2 + 27(u-3)

Distr

Distribute 27

u^4-3u^3 + 9u^2-27u + 3u^3-9u^2 + 27u -81

AddSubTerms

Add and subtract terms

u^4-81

After applying the Distributive Property and simplifying, the result is the same as the given expression. Therefore, we can be sure our solution is correct!

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Exercises p.60-63

Exercise 5 Page 60

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