Exercise 41 Page 61

To factor the given expression we will start by identifying and factoring out the greatest common factor (GCF). Then we will use factoring by grouping and the formula for factoring the difference of perfect 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.

Factoring by Grouping

Now, to factor by grouping, we will take the greatest common factor of the first pair of terms and the greatest common factor of the second pair of terms.

2( 6a^3 + a^2-96a-16)

FactorOut

Factor out a^2

2(a^2( 6a+1 ) - 96a - 16)

FactorOut

Factor out - 16

2(a^2( 6a+1 ) -16( 6a+1 ))

Notice that (6a+1) is a factor of both terms, so we can factor it out.

2(a^2( 6a+1 ) -16( 6a+1 ))

FactorOut

Factor out ( 6a+1 )

2( a^2-16 ) ( 6a+1 )

Difference of Squares

Now, look closely at the expression a^2-16. It can be expressed as the difference of two perfect squares.

a^2-16

WritePow

Write as a power

a^2-4^2

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( a^2- 4^2 )(6a+1) ⇕ 2( a+ 4)( a- 4)(6a+1)

Checking Our Answer

Check your answer ✓

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

2(a + 4)(a - 4) (6a + 1)

Distr

Distribute (a+4)

2(a(a+4) - 4(a+4)) (6a + 1)

▼

Distribute a & - 4

Distr

Distribute a

2(a^2 + 4a - 4(a+4)) (6a + 1)

Distr

Distribute - 4

2(a^2 + 4a - 4a - 16) (6a + 1)

SubTerm

Subtract term

2(a^2 - 16) (6a + 1)

Distr

Distribute (6a+1)

2(a^2(6a + 1) - 16(6a + 1))

▼

Distribute a^2 & -16

Distr

Distribute a^2

2(6a^3 + a^2 - 16(6a + 1))

Distr

Distribute - 16

2(6a^3 + a^2 - 96a - 16)

Distr

Distribute 2

12a^3+2a^2-192a-32

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!