Which model represents the factors of x^2 + 9x + 8? x + 1 x + 8
The Correct Answer and Explanation is:
The correct factors of the quadratic expression x2+9x+8x^2 + 9x + 8 are: (x+1)(x+8)(x + 1)(x + 8)
Explanation
To factor the quadratic expression x2+9x+8x^2 + 9x + 8, we need to break it into two binomials of the form (x+a)(x+b)(x + a)(x + b), such that:
- The product of aa and bb is equal to the constant term 88,
- The sum of aa and bb is equal to the coefficient of the middle term 99.
So we’re looking for two numbers that multiply to 8 and add up to 9.
Let’s list the factor pairs of 8:
- 1×8=81 \times 8 = 8, and 1+8=91 + 8 = 9
- 2×4=82 \times 4 = 8, but 2+4=62 + 4 = 6
Among these, only 1 and 8 satisfy both conditions:
- 1×8=81 \times 8 = 8
- 1+8=91 + 8 = 9
This tells us the correct binomial factors are: (x+1)(x+8)(x + 1)(x + 8)
Let’s verify the factorization by expanding: (x+1)(x+8)=x2+8x+1x+8=x2+9x+8(x + 1)(x + 8) = x^2 + 8x + 1x + 8 = x^2 + 9x + 8
The original expression x2+9x+8x^2 + 9x + 8 is recovered, confirming that the factorization is correct.
This process uses factoring by inspection, a common method when the leading coefficient (the number in front of x2x^2) is 1. This approach involves finding two numbers whose product equals the constant term and whose sum equals the linear coefficient. It’s efficient and especially useful for simple trinomials like this one.
Therefore, the expression x2+9x+8x^2 + 9x + 8 is correctly modeled by: (x+1)(x+8)(x + 1)(x + 8)
