Question

1 multiple choice 20 points
which of the following is the correct factorization of ( x^2 - 16 )?
( (x + 4)^2 )
( (x + 8)(x - 2) )
( (x + 4)(x - 4) )
( (x - 8)(x + 2) )

2 multiple choice 20 points
what is the factored form of ( x^2 + 7x + 10 )?
( (x + 2)(x + 5) )
( (x - 2)(x - 5) )
( (x + 10)(x - 1) )
( (x + 1)(x + 10) )

Explanation:

Question 1

Step1: Recall difference of squares formula

The difference of squares formula is \(a^2 - b^2=(a + b)(a - b)\). For \(x^2-16\), we can rewrite it as \(x^2 - 4^2\), where \(a = x\) and \(b = 4\).

Step2: Apply the formula

Using the difference of squares formula, \(x^2-4^2=(x + 4)(x - 4)\). Now we check the other options:

• \((x + 4)^2=x^2+8x + 16

eq x^2-16\)

• \((x + 8)(x - 2)=x^2+6x-16

eq x^2-16\)

• \((x - 8)(x + 2)=x^2-6x - 16

eq x^2-16\)

Step1: Recall factoring trinomials of the form \(x^2+bx + c\)

We need to find two numbers that multiply to \(c = 10\) and add up to \(b = 7\).

Step2: Find the two numbers

The factors of \(10\) are \(1\) and \(10\), \(2\) and \(5\). Among these, \(2+5 = 7\). So the factored form of \(x^2+7x + 10\) is \((x + 2)(x + 5)\). We check the other options:

• \((x - 2)(x - 5)=x^2-7x + 10

eq x^2+7x + 10\)

• \((x + 10)(x - 1)=x^2+9x-10

eq x^2+7x + 10\)

• \((x + 1)(x + 10)=x^2+11x + 10

eq x^2+7x + 10\)

Answer:

\((x + 4)(x - 4)\)

Question 2