Which of the Following Is a Factor of X ² −6X −16?

Which of the following is a Factor of x ² −6x −16?

A. x + 8

B. 8 - x

C. x - 2

D. x - 6

The correct answer is: B. 8 - x

Here's why

To determine which of the given options is a factor of x^2 - 6x - 16, we can use the factor theorem. According to the theorem, if (x - a) is a factor of a polynomial, then the polynomial equals zero when x = a.

So, let's check each option by substituting the values:

A. x + 8 (x + 8) = 0 x = -8 (-8)^2 - 6(-8) - 16 = 64 + 48 - 16 = 96 ≠ 0 So, x + 8 is not a factor.

B. 8 - x (8 - x) = 0 x = 8 (8)^2 - 6(8) - 16 = 64 - 48 - 16 = 0 So, 8 - x is a factor.

C. x - 2 (x - 2) = 0 x = 2 (2)^2 - 6(2) - 16 = 4 - 12 - 16 = -24 ≠ 0 So, x - 2 is not a factor.

D. x - 6 (x - 6) = 0 x = 6 (6)^2 - 6(6) - 16 = 36 - 36 - 16 = -16 ≠ 0 So, x - 6 is not a factor.

Thus, the correct answer is:

B. 8 - x

Factorisation Using Algebraic Identities

Understanding Algebraic Identities:

• Identities are equations that hold true for all values of the variables involved. They're crucial tools for factoring expressions by rewriting them as products of simpler factors.
• Some common identities include:
  • Square of a Sum: (a + b)^2 = a^2 + 2ab + b^2
  • Square of a Difference: (a - b)^2 = a^2 - 2ab + b^2
  • Difference of Squares: a^2 - b^2 = (a + b)(a - b)
  • Sum-Product Pattern: x^2 + (a + b)x + ab = (x + a)(x + b)

Steps for Factorisation:

1. Examine the expression: Identify the terms, variables, and any potential patterns or structures. Look for squares, differences of squares, or sum-product patterns.
2. Choose an appropriate identity: Based on the structure, apply the relevant identity to rewrite the expression.
3. Simplify the factors: Expand the parentheses and combine like terms if necessary.
4. Check for further factorization: See if the resulting factors can be further broken down using common factors or other identities.

Examples:

1. Factor: x^2 + 6x + 9

• This resembles a perfect square trinomial with a = x and b = 3.
• Apply the "Square of a Sum" identity: (x + 3)^2

2. Factor: a^2 - 49

• This is a difference of squares with a = a and b = 7.
• Apply the "Difference of Squares" identity: (a + 7)(a - 7)

3. Factor: 3x^2 + 7x + 2

• This doesn't directly match a specific identity, but we can rewrite the second term (7x) to fit the "Sum-Product Pattern."
• Rewrite as 3x^2 + 6x + x + 2.
• Group terms: (3x^2 + 6x) + (x + 2).
• Factor out common factors: 3x(x + 2) + 1(x + 2).
• Combine factors: (3x + 1)(x + 2).