Exercise 6 Page 60

To factor the given expression, we will first identify and factor out the greatest common factor. Then, we will use the formula for the difference of two squares.

Factor Out the GCF

The greatest common factor (GCF) of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. The GCF of the given expression is 2.

Difference of Squares

Look closely at the expression d^4-16f^4. It can be expressed as the difference of two perfect squares.

d^4-16f^4

WritePow

Write as a power

(d^2)^2-4^2(f^2)^2

ProdPowII

a^m b^m = (a b)^m

(d^2)^2-(4f^2)^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. 2 ( ( d^2)^2-( 4f^2)^2 ) ⇕ 2( d^2+ 4f^2)( d^2- 4f^2) Now, let's take a look at the expression d^2-4f^2. Once again, we can express it as a difference of two perfect squares. Let's do it!

d^2-4f^2

WritePow

Write as a power

d^2-2^2f^2

ProdPowII

a^m b^m = (a b)^m

d^2-(2f)^2

Now, we can apply the above formula to factor our expression completely. 2(d^2+4f^2)( d^2-( 2f)^2) ⇕ 2(d^2+4f^2)( d+ 2f)( d- 2f)

Checking Our Answer

Check your answer ✓

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

2 (d^2 + 4f^2) (d + 2f)(d-2f)

Distr

Distribute (d+2f)

2(d^2 + 4f^2) (d(d+2f) - 2f(d+2f))

▼

Distribute d & -2f

Distr

Distribute d

2(d^2 + 4f^2) (d^2+2df - 2f(d+2f))

Distr

Distribute - 2f

2(d^2 + 4f^2) (d^2+2df - 2df-4f^2)

SubTerm

Subtract term

2(d^2 + 4f^2) (d^2-4f^2)

Distr

Distribute (d^2+4f^2)

2(d^2(d^2+4f^2)-4f^2(d^2+4f^2))

▼

Distribute d^2 & - 4f^2

Distr

Distribute d^2

2(d^4+4d^2f^2-4f^2(d^2+4f^2))

Distr

Distribute - 4f^2

2(d^4+4d^2f^2-4d^2f^2-16f^4)

SubTerm

Subtract term

2(d^4-16f^4)

Distr

Distribute 2

2d^4-32f^4

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!