Question

rewrite as binomials then multiply using foil.
1
two less than a number is multiplied by three more than the same number.
2
a number decreased by six is multiplied by the same number decreased by twelve.

Explanation:

Problem 1

Step1: Define the number

Let the number be \( x \). "Two less than a number" is \( x - 2 \), and "three more than the same number" is \( x + 3 \). So the binomials are \( (x - 2) \) and \( (x + 3) \).

Step2: Apply FOIL method

FOIL stands for First, Outer, Inner, Last.

• First: \( x \times x = x^2 \)
• Outer: \( x \times 3 = 3x \)
• Inner: \( -2 \times x = -2x \)
• Last: \( -2 \times 3 = -6 \)

Now, combine like terms: \( x^2 + 3x - 2x - 6 = x^2 + x - 6 \)

Step1: Define the number

Let the number be \( x \). "A number decreased by six" is \( x - 6 \), and "the same number decreased by twelve" is \( x - 12 \). So the binomials are \( (x - 6) \) and \( (x - 12) \).

Step2: Apply FOIL method

• First: \( x \times x = x^2 \)
• Outer: \( x \times (-12) = -12x \)
• Inner: \( -6 \times x = -6x \)
• Last: \( -6 \times (-12) = 72 \)

Now, combine like terms: \( x^2 - 12x - 6x + 72 = x^2 - 18x + 72 \)

Answer:

The product is \( x^2 + x - 6 \)

Problem 2